Alexander scored twice in another impressive showing from Burnley
Burnley increased the pressure on Hull City manager Phil Brown after the game turned on two controversial decisions.
Burnley's opener came from a Graham Alexander penalty after referee Mike Jones surprisingly ruled that Tyrone Mears had been fouled by Stephen Hunt.
Hull's Geovanni appeared to equalise direct from a free-kick, but Jones disallowed it for pushing in the wall.
The aggrieved Geovanni quickly picked up two bookings to be sent off and Alexander's second goal settled it.
First, the Brazilian was booked for remonstrating long and hard with Jones after his right-footed free-kick, just outside the area, had crept into the top corner via Burnley keeper Brian Jensen.
Jones - who had officiated when Liverpool lost to the infamous "beach ball" goal scored by Sunderland's Darren Bent - had blown his whistle having seen Seyi Olofinjana push Wade Elliott in the wall.
The luck will change - Brown
But Hull's players clearly felt it was an extremely harsh decision.
Geovanni was then shown a second yellow card for a wild lunge at Steven Fletcher and with time running out - and Hull increasingly stretched - Alexander drilled a shot from outside of the area into the bottom corner.
With Adam Pearson - expected to be named as Hull's new chairman on Monday - watching the hapless display as he sat beside majority shareholder Russell Bartlett in the directors' box, the pressure was squarely on Brown.
After a scrappy opening, the first shot of the match took 15 minutes in arriving, with Olofinjana forcing Jensen into a straightforward save following good work from Kamel Ghilas.
It did allow Jensen to test the ankle he had twisted last weekend, with the Clarets playing the same side beaten at home by Wigan.
The announcement will be forthwith and hopefully I'll be standing alongside Adam Pearson on Monday
Hull, meanwhile, replaced Bernard Mendy with midfielder Ghilas, positioned wide on the right. Brown had been forced into a second change, with regular keeper Boaz Myhill out injured and Matt Duke taking his place.
Duke enjoyed a quiet start - until referee Jones awarded an extremely dubious penalty in the 19th minute.
Released by Wade Elliott, and well positioned goal-side of Hunt, Mears made a poor attempt at a shot and fell in a heap. But Jones decided Hunt had fouled Elliott, and swiftly pointed to the spot - ignoring the howls of protest from Hull's players.
After Alexander had slotted home his penalty, beating Duke at his right-hand post, Hull almost found themselves 2-0 down - but Kamil Zayatte did well to block a fierce shot from Robbie Blake.
Burnley continued to look the more threatening of the two sides, with Andre Bikey's header flashing over the bar, and Elliott forcing Duke to save smartly from close range.
Duke became one of the busiest players on the park in the lead-up to half-time, with a beautiful Burnley move rounded off by a well-placed strike from Chris Eagles that Duke saved - though the rebound only narrowly evaded the prowling Elliott.
After half-time, Hull created a wonderful opportunity triggered by a beautiful pass from Jan Vennegoor of Hesselink, who had spent the first half wasted as an isolated striker.
The Dutchman's angled ball into the penalty area found Olofinjana, and he in turn set up Ghilas, but the Algerian - with only Jensen to beat, fired his shot straight at the big Dane.
A massive three points for us - Coyle
As precious minutes ticked by without any further openings, Brown elected to bring off Vennegoor of Hesselink and Paul McShane, the latter reacting furiously by kicking over a crate of drinks bottles.
Mendy and Jozy Altidore were the men introduced for Hull, who had barely half an hour to get something out of the game.
A purposeful run and shot by Mendy did not test Jensen sufficiently, however, and - although Hull then seemed to have got onto the scoresheet through Geovanni's free-kick - it proved an ultimately fruitless afternoon for the visitors.
Brown, widely reported to need all three points from this encounter to avert a sacking, remained impassive through the moments of controversy and in the immediate aftermath of the match.
But with Burnley registering their fifth home win of the season, and his own side deeply mired in the relegation zone - it could be a matter of time before he is replaced as Hull manager.
Conversely, the Clarets, widely expected to struggle on their long-awaited return to the top flight, are sitting pretty in mid-table.
Burnley manager Owen Coyle: "There was a clamour for changes here but I showed faith with the same 11 men - because we played well against Wigan, although we lost the game - and Graham Alexander came up trumps again.
"He trains every session as though it was his last, I am delighted that he keeps producing the goods for me.
"In terms of finance it's a total mismatch. My players are the lowest paid in the Premier League but they belie that."
"There was a big decision with the disallowed goal but I think one of their players pulled one of our lads out of the road, though I don't think it affected the flight of the ball.
"[Phil Brown] is a personal friend of mine. You don't like to see anyone down on their luck. We are friends and we go back a long way. I am sure he will see this through and come back stronger."
90:00+4:12 The referee blows his whistle to end the game.
90:00+2:45 Kamil Zayatte gives away a free kick for an unfair challenge on Kevin McDonald. Wade Elliott takes the direct free kick.
90:00+1:10 Free kick awarded for a foul by Bernard Mendy on Fernando Guerrero. Fernando Guerrero takes the free kick.
89:23 Booking The referee books Bernard Mendy for unsporting behaviour.
88:31 Free kick awarded for a foul by Stephen Jordan on Bernard Mendy. Free kick crossed left-footed by Andy Dawson, save made by
Brian Jensen. Kamel Ghilas takes the inswinging corner, David Nugent makes a clearance. Andy Dawson produces a left-footed
shot from just outside the area that bounces off the post.
88:08 Kevin McDonald has a mishit shot. Kamil Zayatte gets a block in.
87:18 Effort from deep inside the area by Andre Bikey clears the bar.
85:17 Jozy Altidore fouled by Steven Caldwell, the ref awards a free kick. Andy Dawson crosses the ball from the free kick left-footed
from right wing, save made by Brian Jensen.
83:50 Matt Duke takes the direct free kick.
83:50 Substitution Fernando Guerrero comes on in place of Chris Eagles.
83:50 Unfair challenge on Nick Barmby by David Nugent results in a free kick.
83:38 A cross is delivered by Chris Eagles.
81:37 Jozy Altidore challenges Tyrone Mears unfairly and gives away a free kick. The free kick is swung in right-footed by Chris
Eagles, Kamil Zayatte manages to make a clearance.
81:15 Seyi Olofinjana takes a shot. Steven Caldwell gets a block in.
79:52 Inswinging corner taken from the left by-line by Chris Eagles.
79:04 Effort on goal by Tyrone Mears from inside the area goes harmlessly over the bar.
76:35 Substitution (Burnley) makes a substitution, with David Nugent coming on for Steven Fletcher.
76:35 The assist for the goal came from Kevin McDonald.
76:35 GOAL - Graham Alexander:Burnley 2 - 0 Hull Graham Alexander grabs a goal from just outside the penalty box to the bottom left corner of the goal. Burnley 2-0 Hull.
75:13 Free kick awarded for an unfair challenge on Chris Eagles by Seyi Olofinjana. Chris Eagles crosses the ball from the free
kick right-footed from right wing, Andy Dawson manages to make a clearance.
71:58 Substitution Kevin McDonald comes on in place of Robbie Blake.
69:58 Substitution Stephen Hunt leaves the field to be replaced by Nick Barmby.
69:30 Brian Jensen restarts play with the free kick.
69:30 Sent off Deiberson Geovanni receives a red card.
69:30 Sent off Deiberson Geovanni receives a red card.
Correction - 69:30 Direct free kick taken by Brian Jensen.
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.