Conway (right) scored the second goal against St Mirren
Dundee United stayed in touch at the top of the Scottish Premier League despite being reduced to 10 men with the sending off of Morgaro Gomis.
Damian Casalinuovo fired the ball into the far corner after 29 minutes and Craig Conway's free-kick deceived keeper Paul Gallacher on 44 minutes.
Gomis was red carded and Darren Dods headed an own goal from the free-kick.
Jennison Myrie-Williams restored the lead with a neat chip, but Stephen O'Donnell replied from close range.
It meant a nervous finish for United, but they held out to extend their unbeaten league run to eight matches.
Defender Sean Dillon had been drafted into United's midfield in place of the suspended Prince Buaben, while David Goodwillie replaced Danny Cadamarteri in attack.
But the big surprise was a place on the bench for Scott Robertson, who has been out for two months with a groin problem.
St Mirren's only change saw striker Michael Higdon come in for Andy Dorman after the midfielder was ruled out for up to six weeks with a hamstring injury.
But it was St Mirren who dominated possession early on and Hugh Murray volleyed over a waist-high Jack Ross cross after 10 minutes.
The visitors were then on the wrong end of two marginal offside decisions inside two minutes.
The first did not matter as Billy Mehmet shanked his finish, but the second saw strike partner Higdon net from Garry Brady's poke through.
The tame atmosphere on and off the field was shaken when Stephen McGinn took out Gomis in mid-air 21 minutes in and both sets of players had to be separated as the St Mirren midfielder was booked.
Higdon then missed the best chance so far when he turned Murray's cross wide from close range.
St Mirren's Garry Brady and United's Jennison Myrie-Williams had claims for penalties turned down after going to ground theatrically.
There was now an edge to the game and a goal soon followed after 29 minutes when when a perfectly weighted through-ball from Dillon allowed Casalinuovo to turn in the box, shrug off Chris Innes and side-foot past Gallacher.
Saints should have been 2-0 down seven minutes later when Dillon once again picked out Casalinuovo in the box and the Argentinian's knockdown was perfect for Goodwillie, but the striker made a hash of his finish from six yards.
Casalinuovo opened the scoring for Dundee United
However, two minutes before the break, Gallacher was left red faced when Conway curled in a free-kick after the keeper left a huge gap at his near post.
United appeared to be coasting, but the game turned in the 59th minute when Gomis was sent off for what appeared to be a slap on opposing midfielder Stevie Thomson.
And, from the resulting free-kick, Dods seemed to get the last touch to the ball under pressure from Lee Mair.
United were in danger of throwing the game away, although St Mirren were not doing themselves any favours by passing the ball more poorly than when they did not have the extra man.
However, Conway played the ball through to Myrie-Williams, who lifted the ball beautifully over Gallacher.
With United down to nine men with Dods off the field with a blood injury, O'Donnell turned the ball in with five minutes remaining.
Dundee United manager Craig Levein:
"I now need to address the red cards we are getting, but I would not say we are, in any terms, a dirty side.
"I think four of the six red cards have not been red cards, but we can't allow it to continue.
"Morgaro is an honest boy and he tells me it's retaliation: he was elbowed in the face and he slapped him across the back of the head and the ref only sees that.
"A few things went against us, but credit to the boys, it was a fantastic match, we played some good stuff and we won the game, which is the most important thing."
St Mirren manager Gus MacPherson:
"We can't keep coming out after games and saying we are playing well and losing the goals we are losing.
"For 30 minutes, we were controlling the game and were comfortable.
"We've been criticised in the past for being stuffy, defensive and in the last 18 months we have made a real effort to go and play football.
"You still have to have a platform at the back, but we don't have that and slackness at the back has cost us."
90:00+0:37 Centre by Conor Ramsey, clearance by Andy Webster.
88:35 Chris Innes challenges Craig Conway unfairly and gives away a free kick. Garry Kenneth takes the direct free kick.
87:26 Nicky Weaver takes the direct free kick.
87:26 Booking Stephen O'Donnell goes into the book.
87:16 Stephen O'Donnell concedes a free kick for a foul on Paul Dixon.
85:48 Stephen O'Donnell takes a inswinging corner from the left by-line to the near post, clearance by Andy Webster.
85:24 Substitution Garry Kenneth joins the action as a substitute, replacing Darren Dods.
84:30 Jack Ross provided the assist for the goal.
84:30 GOAL - Stephen O'Donnell:Dundee Utd 3 - 2 St Mirren Stephen O'Donnell scores a goal from deep inside the penalty area to the bottom right corner of the goal. Dundee Utd 3-2 St
83:25 Craig Conway produces a right-footed shot from the edge of the box and misses to the left of the target.
82:37 Free kick awarded for an unfair challenge on Andy Webster by Conor Ramsey. Free kick taken by Nicky Weaver.
81:24 Free kick awarded for an unfair challenge on Stephen McGinn by Sean Dillon. Lee Mair takes the direct free kick.
80:11 Direct free kick taken by Andy Webster.
80:11 Substitution Conor Ramsey on for Steven Thomson.
80:11 Booking Chris Innes receives a yellow card for unsporting behaviour.
80:03 Chris Innes challenges Craig Conway unfairly and gives away a free kick.
79:55 The ball is swung over by David Barron, Andy Webster manages to make a clearance.
79:36 The ball is crossed by Jack Ross, clearance by Michael Kovacevic.
78:18 Craig Conway provided the assist for the goal.
78:18 GOAL - Jennison Myrie-Williams:Dundee Utd 3 - 1 St Mirren A goal is scored by Jennison Myrie-Williams from close range to the bottom right corner of the goal. Dundee Utd 3-1 St Mirren.
77:26 Billy Mehmet challenges Paul Dixon unfairly and gives away a free kick. Direct free kick taken by Paul Dixon.
76:51 Jennison Myrie-Williams delivers the ball.
76:04 Free kick taken by Andy Webster.
76:04 Booking Booking for David Barron.
75:53 David Barron challenges Jennison Myrie-Williams unfairly and gives away a free kick.
75:22 Substitution (St Mirren) makes a substitution, with Steven Robb coming on for Hugh Murray.
75:07 Shot by Stephen McGinn from outside the box goes high over the target.
74:52 Inswinging corner taken right-footed by Stephen O'Donnell from the left by-line, clearance made by Darren Dods.
Correction - 74:26 Stephen O'Donnell produces a cross, blocked by Michael Kovacevic.
74:26 A cross is delivered by Stephen O'Donnell, blocked by Michael Kovacevic.
73:15 Centre by Stephen O'Donnell, clearance made by Darren Dods.
72:26 Billy Mehmet is penalised for a handball. Nicky Weaver takes the indirect free kick.
71:46 The ball is sent over by David Barron.
71:25 The ball is crossed by Hugh Murray, clearance made by Scott Robertson.
70:22 The assistant referee flags for offside against Stephen O'Donnell. Nicky Weaver takes the free kick.
67:48 Steven Thomson concedes a free kick for a foul on Jon Daly. Direct free kick taken by Sean Dillon.
66:01 The ball is delivered by Paul Dixon.
65:30 The referee blows for offside against Michael Higdon. Nicky Weaver restarts play with the free kick.
64:41 Substitution Jon Daly on for Damian Casalinuovo.
64:41 Substitution Stephen O'Donnell is brought on as a substitute for Garry Brady.
62:44 Substitution Scott Robertson on for David Goodwillie.
60:50 GOAL - Darren Dods:Dundee Utd 2 - 1 St Mirren Headed own goal by Darren Dods. Dundee Utd 2-1 St Mirren.
59:53 Jack Ross delivers the ball from the free kick right-footed from right channel, blocked by Darren Dods.
58:23 Sent off Morgaro Gomis is sent off by the referee for violent conduct.
57:10 The ball is swung over by Jack Ross, clearance made by Darren Dods.
56:34 Free kick awarded for an unfair challenge on Craig Conway by Steven Thomson. Nicky Weaver restarts play with the free kick.
54:55 Direct effort from the free kick comes in from Jack Ross.
54:55 Booking Jennison Myrie-Williams receives a yellow card for unsporting behaviour.
54:48 Steven Thomson fouled by Jennison Myrie-Williams, the ref awards a free kick.
54:25 Corner taken right-footed by Garry Brady from the left by-line, Paul Dixon manages to make a clearance.
54:16 Billy Mehmet sends in a cross, Darren Dods manages to make a clearance.
52:22 Darren Dods fouled by Billy Mehmet, the ref awards a free kick. Direct free kick taken by Nicky Weaver.
51:29 David Barron takes the free kick.
51:29 Booking David Goodwillie is booked.
51:23 Foul by David Goodwillie on Garry Brady, free kick awarded.
51:09 Morgaro Gomis crosses the ball, clearance by Lee Mair.
49:47 David Barron concedes a free kick for a foul on Jennison Myrie-Williams. Nicky Weaver takes the direct free kick.
47:30 Paul Dixon produces a cross, Damian Casalinuovo has an effort at goal from close in which goes wide left of the goal.
47:14 Morgaro Gomis fouled by Hugh Murray, the ref awards a free kick. Paul Dixon restarts play with the free kick.
45:01 The second half kicks off.
45:00+1:36 The ref blows to signal half-time.
45:00+0:39 Jennison Myrie-Williams concedes a free kick for a foul on David Barron. Paul Gallacher restarts play with the free kick.
45:00+0:27 Lee Mair gives away a free kick for an unfair challenge on David Goodwillie. Free kick taken by Morgaro Gomis.
44:46 Billy Mehmet produces a curled right-footed shot from close in and bounces off the post.
43:44 GOAL - Craig Conway:Dundee Utd 2 - 0 St Mirren Craig Conway finds the net with a goal direct from the free kick from outside the penalty box to the top left corner of the
goal. Dundee Utd 2-0 St Mirren.
43:21 Hugh Murray is penalised for a handball.
42:32 Craig Conway takes a shot. Jack Ross gets a block in.
41:32 Darren Dods challenges Billy Mehmet unfairly and gives away a free kick. Free kick crossed right-footed by Jack Ross, clearance
by Darren Dods.
40:53 Steven Thomson concedes a free kick for a foul on Sean Dillon. Nicky Weaver takes the free kick.
37:00 Damian Casalinuovo handles the ball and concedes a free kick. Chris Innes takes the direct free kick.
36:08 David Goodwillie takes a shot.
34:53 Garry Brady takes a corner.
34:33 Stephen McGinn takes a shot. Sean Dillon gets a block in.
33:35 Steven Thomson challenges David Goodwillie unfairly and gives away a free kick. Direct free kick taken by Paul Dixon.
32:37 The ball is sent over by Sean Dillon.
32:18 Corner taken by Craig Conway from the left by-line, Michael Higdon makes a clearance.
32:09 Morgaro Gomis produces a cross, clearance by Chris Innes.
31:32 Inswinging corner taken right-footed by Craig Conway, save made by Paul Gallacher.
30:44 David Barron crosses the ball, Darren Dods makes a clearance.
28:53 Assist on the goal came from Sean Dillon.
28:53 GOAL - Damian Casalinuovo:Dundee Utd 1 - 0 St Mirren Damian Casalinuovo finds the back of the net with a goal from inside the six-yard box to the bottom left corner of the goal.
Dundee Utd 1-0 St Mirren.
26:37 Steven Thomson challenges Craig Conway unfairly and gives away a free kick. Free kick crossed right-footed by Craig Conway,
clearance made by Michael Higdon.
22:58 The ball is delivered by Hugh Murray, Effort from inside the six-yard box by Michael Higdon goes wide right of the target.
22:25 Damian Casalinuovo gives away a free kick for an unfair challenge on Chris Innes. Paul Gallacher takes the direct free kick.
21:12 Direct free kick taken by Nicky Weaver.
21:12 Booking Stephen McGinn goes into the book for unsporting behaviour.
21:04 Stephen McGinn challenges Morgaro Gomis unfairly and gives away a free kick.
20:45 Free kick awarded for an unfair challenge on Garry Brady by Jennison Myrie-Williams. Jack Ross crosses the ball from the free
kick right-footed from left wing, Andy Webster manages to make a clearance.
18:30 Inswinging corner taken right-footed by Craig Conway, save by Paul Gallacher.
18:21 Damian Casalinuovo crosses the ball, Chris Innes gets a block in.
17:06 Unfair challenge on Jennison Myrie-Williams by David Barron results in a free kick. Direct free kick taken by Andy Webster.
15:02 A cross is delivered by Michael Higdon, clearance by Andy Webster.
13:31 Damian Casalinuovo gives away a free kick for an unfair challenge on Chris Innes. Free kick taken by Paul Gallacher.
13:10 The assistant referee signals for offside against Michael Higdon. Free kick taken by Nicky Weaver.
13:01 Centre by Jack Ross.
11:46 Shot from long range by Craig Conway goes wide of the left-hand upright.
11:00 The assistant referee flags for offside against Billy Mehmet. Indirect free kick taken by Nicky Weaver.
9:41 Jack Ross sends in a cross, Shot on goal by Steven Thomson from just inside the area clears the crossbar.
8:48 Shot by David Goodwillie went wide of the right-hand post.
7:42 Craig Conway crosses the ball, David Barron manages to make a clearance.
5:30 Steven Thomson takes a shot. Save made by Nicky Weaver.
4:13 Free kick awarded for a foul by Damian Casalinuovo on Chris Innes. Paul Gallacher takes the free kick.
3:51 The assistant referee flags for offside against Billy Mehmet. Free kick taken by Nicky Weaver.
2:54 The ball is swung over by Jack Ross, Nicky Weaver makes a save.
2:32 Jack Ross produces a cross, Darren Dods makes a clearance.
0:56 Paul Dixon crosses the ball.
0:00 The referee gets the game underway.
Live text and data provided by The Press Association.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.