Wade Elliott celebrates after scoring the first top-flight goal of his career
Burnley followed up their win over Manchester United by beating Everton after Louis Saha missed a penalty.
A storming start by the Clarets saw Martin Paterson head Chris McCann's cross against the bar after 40 seconds.
Wade Elliott gave Burnley a deserved lead with a neat finish that took a slight deflection off Phil Neville.
Everton improved after the break and their best chance to level came when Tony Hibbert went down in the area but Saha fired his spot-kick wide.
The Toffees are left without a point to show for their first two Premier League matches and their boss David Moyes has much to ponder before the transfer window slams shut in nine days' time.
True, this was nothing like as painful an experience as Everton's opening-day humiliation at the hands of Arsenal but it still exposed the limitations in their current line-up.
The departure of want-away defender Joleon Lescott to Manchester City cannot happen soon enough for Moyes - to end the disruption caused by such a prolonged transfer saga and, just as importantly, allow him to spend the proceeds on improving his squad.
Everton are short of numbers in defence, where Neville again had to fill in at centre-back, but they are also short of ideas going forward and lone striker Louis Saha had only scraps to feed on up front at Turf Moor.
Burnley, in contrast, were full of confidence after beating the Premier League champions five days ago and they could have taken the lead inside the first minute.
McCann crossed from the left and Paterson headed against the bar before Steven Fletcher, who was following up, forced Tim Howard into a low save.
The Everton defence looked uncomfortable in the early stages and they were soon exposed again when Elliott made space for a cross from the right and Paterson headed wide when he should really have found the target.
We deserved the victory - Coyle
And, while the visitors eventually managed to plug those gaps at the back, they took a great deal longer to threaten at the other end.
Everton's first meaningful effort came after 30 minutes and saw Jack Rodwell head Leighton Baines' corner straight at Brian Jensen.
Seconds later, Tony Hibbert's low cross was just too strong for Saha at the back post but, having just had their best spell of the match, Everton found themselves behind.
After a dangerous Robbie Blake cross had fizzed across the face of goal, Steven Fletcher teed up Elliott, who composed himself before coolly curling the ball into the corner of the net, aided by a touch from Neville's boot.
Everton did come out with more purpose after the break, no doubt fired up by angry words from Moyes.
But Burnley were still playing the neater football and went close to extending their lead when the quick-thinking McCann saw his attempted lob tipped over by Howard after the ball broke to him in the box.
We could do with new players - Moyes
The Toffees responded by piling on the pressure and they should have equalised when Phil Dowd awarded them a disputed penalty with 15 minutes left.
Hibbert burst into the box and went to ground even though Chris McCann had pulled out of the tackle and made minimal, if any, contact.
Dowd was convinced it was a foul, however, but it made no difference to the final outcome as Saha sent the ball wide of Jensen's left-hand post.
The Danish goalkeeper had been one of the heroes of Wednesday's win over Manchester United, saving Michael Carrick's spot-kick, but he was not called upon to make another stop in the closing stages on this occasion.
Saha almost made amends for his penalty miss by drilling a rasping 30-yard shot inches wide but, despite Everton's late dominance, Burnley were able to hang on for the points.
Burnley boss Owen Coyle: "It is another three points but that is all it is.
"We've had a great week but that is all it has been. There is so much more ahead of us, so much hard work.
"I felt over the course of the game we were the better team. I don't think they cut us open too often."
Everton boss David Moyes: "Burnley were the better side in the opening 25 minutes. I think, with the confidence they gained from the game in midweek, we expected that.
"But after that I thought we got back into it and were a bit unfortunate not to get something out of it."
90:00+4:16 The referee blows his whistle to end the game.
Correction - 90:00+1:49 Unfair challenge on Louis Saha by Andre Bikey results in a free kick. Joao Alves Jo fires a strike on goal direct from the
free kick, Steven Thompson gets a block in. Headed effort from deep inside the area by Jack Rodwell misses to the right of
90:00+1:49 Unfair challenge on Louis Saha by Andre Bikey results in a free kick. Strike on goal comes in from Joao Alves Jo from the
free kick, blocked by Steven Thompson. Header by Jack Rodwell from deep inside the penalty area misses to the right of the
90:00+0:33 Chris Eagles fouled by Phil Neville, the ref awards a free kick. Direct free kick taken by Chris Eagles.
89:37 Free kick awarded for a foul by Joao Alves Jo on Clarke Carlisle. Direct free kick taken by Brian Jensen.
88:48 Centre by Fernando Guerrero.
86:50 Inswinging corner taken from the right by-line by Leighton Baines, save by Brian Jensen. Steven Pienaar crosses the ball,
Close range header by Joao Alves Jo misses to the right of the target.
86:39 Jack Rodwell takes a shot. Blocked by Wade Elliott.
85:33 Substitution Steven Thompson joins the action as a substitute, replacing Steven Fletcher.
84:45 Tyrone Mears sends in a cross, save by Tim Howard.
83:31 Substitution Fernando Guerrero on for Robbie Blake.
81:55 Unfair challenge on Joseph Yobo by Steven Fletcher results in a free kick. Tim Howard restarts play with the free kick.
81:23 Leon Osman concedes a free kick for a foul on Stephen Jordan. Free kick taken by Brian Jensen.
80:20 Unfair challenge on Graham Alexander by Leon Osman results in a free kick. Direct free kick taken by Brian Jensen.
80:12 Centre by Tony Hibbert, Stephen Jordan gets a block in.
79:17 Effort from outside the penalty box by Louis Saha goes wide right of the target.
78:42 Substitution Chris Eagles joins the action as a substitute, replacing Martin Paterson.
77:06 Shot by Joao Alves Jo from 18 yards. Blocked by Andre Bikey. Leighton Baines takes a corner.
75:57 Joao Alves Jo takes a shot. Blocked by Graham Alexander. Corner from the left by-line taken by Leighton Baines, Louis Saha
concedes a free kick for a foul on Clarke Carlisle. Free kick taken by Brian Jensen.
74:13 Penalty missed by Louis Saha.
74:13 Penalty awarded for an unfair challenge on Tony Hibbert by Chris McCann.
73:12 Foul by Steven Fletcher on Steven Pienaar, free kick awarded. Direct free kick taken by Steven Pienaar.
71:43 Drilled right-footed shot by Steven Pienaar. Blocked by Clarke Carlisle.
70:25 Unfair challenge on Leighton Baines by Steven Fletcher results in a free kick. Phil Neville takes the direct free kick.
66:53 Leighton Baines sends in a cross, clearance by Andre Bikey.
66:18 Louis Saha fouled by Graham Alexander, the ref awards a free kick. Free kick taken by Tony Hibbert.
64:48 Direct free kick taken by Clarke Carlisle.
64:48 Substitution Joao Alves Jo comes on in place of Marouane Fellaini.
64:48 Handball by Steven Pienaar.
64:22 The ball is sent over by Tony Hibbert.
63:55 Steven Fletcher has an effort at goal from outside the area which goes wide right of the goal.
63:38 Unfair challenge on Graham Alexander by Marouane Fellaini results in a free kick. Direct free kick taken by Clarke Carlisle.
63:28 Handball decision goes against Martin Paterson. Direct free kick taken by Phil Neville.
62:34 Shot by Chris McCann from 12 yards. Save by Tim Howard. Inswinging corner taken right-footed by Robbie Blake from the left
by-line, Tim Howard makes a save.
62:16 Graham Alexander takes a shot. Blocked by Chris McCann.
61:40 Robbie Blake decides to take the corner short.
60:10 Tim Cahill challenges Steven Fletcher unfairly and gives away a free kick. Free kick crossed right-footed by Robbie Blake,
clearance made by Phil Neville.
59:23 Foul by Tim Cahill on Andre Bikey, free kick awarded. Brian Jensen takes the free kick.
59:14 Leon Osman crosses the ball, clearance made by Andre Bikey.
58:22 Tim Cahill takes a shot. Blocked by Tyrone Mears. Corner taken left-footed by Leighton Baines to the near post, Graham Alexander
manages to make a clearance.
55:36 Louis Saha has an effort at goal from outside the box which misses left.
55:28 Leon Osman crosses the ball, Wade Elliott manages to make a clearance.
54:33 The ball is sent over by Chris McCann.
50:07 Martin Paterson is caught offside. Direct free kick taken by Phil Neville.
48:59 Corner taken right-footed by Robbie Blake, save made by Tim Howard.
47:15 Phil Neville takes the direct free kick.
47:15 Booking Wade Elliott is cautioned by the ref for unsporting behaviour.
47:10 Wade Elliott challenges Marouane Fellaini unfairly and gives away a free kick.
46:14 The referee blows for offside against Steven Pienaar. Indirect free kick taken by Brian Jensen.
45:01 The second half kicks off.
45:00+1:12 The players leave the pitch at half time.
45:00+0:45 The ball is sent over by Steven Pienaar, clearance by Andre Bikey.
45:00+0:29 Leighton Baines takes a shot. Wade Elliott gets a block in.
44:31 The referee blows for offside. Tim Howard takes the indirect free kick.
41:54 Foul by Wade Elliott on Leighton Baines, free kick awarded. Direct free kick taken by Tim Howard.
41:25 Marouane Fellaini gives away a free kick for an unfair challenge on Clarke Carlisle. Direct free kick taken by Brian Jensen.
39:58 Corner taken by Steven Pienaar from the left by-line played to the near post, Brian Jensen makes a save.
38:55 Free kick awarded for a foul by Tim Cahill on Steven Fletcher. Free kick taken by Graham Alexander.
37:56 Tyrone Mears fouled by Louis Saha, the ref awards a free kick. Free kick taken by Brian Jensen.
37:26 Handball by Wade Elliott. Direct free kick taken by Leighton Baines.
Correction - 35:31 Steven Pienaar fouled by Graham Alexander, the ref awards a free kick. Steven Pienaar crosses the ball in from the free kick,
save made by Brian Jensen. Shot from 25 yards by Leon Osman. Robbie Blake gets a block in.
35:31 Free kick awarded for an unfair challenge on Steven Pienaar by Graham Alexander. The free kick is swung in left-footed by
Steven Pienaar, save by Brian Jensen. Andre Bikey takes the chance to get an effort at goal. Robbie Blake gets a block in.
33:54 Assist by Steven Fletcher.
33:54 GOAL - Wade Elliott:Burnley 1 - 0 Everton Wade Elliott fires in a goal from the edge of the penalty box to the bottom left corner of the goal. Burnley 1-0 Everton.
31:58 Tony Hibbert crosses the ball.
29:49 The ball is crossed by Tony Hibbert, clearance made by Tyrone Mears. Outswinging corner taken left-footed by Leighton Baines
from the left by-line, Jack Rodwell takes a shot. Save by Brian Jensen.
27:55 Steven Pienaar has an effort at goal from outside the area missing to the wide right of the target.
25:37 Unfair challenge on Wade Elliott by Marouane Fellaini results in a free kick. Clarke Carlisle restarts play with the free
21:43 Andre Bikey concedes a free kick for a foul on Louis Saha. Jack Rodwell has an effort direct from the free kick.
17:39 The ball is swung over by Chris McCann, Phil Neville makes a clearance. Effort on goal by Steven Fletcher from inside the
box misses to the right of the target.
16:32 The ball is sent over by Leon Osman.
15:29 The ball is delivered by Chris McCann, Tim Howard makes a save.
14:09 Jack Rodwell takes a shot. Blocked by Andre Bikey.
13:08 Clarke Carlisle concedes a free kick for a foul on Louis Saha. Phil Neville takes the direct free kick.
10:09 Leon Osman takes a shot. Blocked by Clarke Carlisle.
9:07 Steven Fletcher delivers the ball, clearance by Jack Rodwell.
7:46 The ball is crossed by Chris McCann.
5:37 Tony Hibbert sends in a cross.
2:54 The ball is crossed by Wade Elliott, Header from close in by Martin Paterson misses to the left of the goal.
1:36 Robbie Blake fouled by Tim Cahill, the ref awards a free kick. Robbie Blake restarts play with the free kick.
1:08 Wade Elliott takes a shot. Save made by Tim Howard.
0:40 The ball is sent over by Chris McCann, Martin Paterson takes a shot.
0:00 The ref blows the whistle to start the match.
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