Rodolph Austin scored his first goal since September as Leeds picked up their second win in four days under new manager Brian McDermott.
Jamaica international Austin latched on to El-Hadji Diouf's throughball and swept the home beyond Lee Grant.
Chris McCann had an effort turned away, and Ross McCormack saw a volley saved.
A second straight Championship win lifts Leeds further away from relegation trouble but moves Burnley to within three points of the drop zone.
Leeds have now won two games in a row since Brian McDermott's appointment after previously losing their last four Championship games
In the absence of leading goalscorer Charlie Austin, following emergency surgery to remove his appendix, the Clarets created little in front of goal.
Leeds twice went close to taking the lead when defender Jason Pearce's header from Stephen Warnock's cross was blocked by Burnley captain Jason Shackell on the line and Luke Varney reacted quickest to head the rebound on to the bar.
Burnley's Martin Paterson lashed an angled drive inches wide after Ross Wallace had drilled a trademark effort from outside the penalty area into the arms of Paddy Kenny.
Varney was then denied by Grant, who tipped his far post header from McCormack's free-kick over the crossbar, before Diouf showed good technique to control McCormack's pass to clip the ball over the top for Austin to side-foot home.
Leeds manager Brian McDermott:
"I've come here to enjoy myself, the staff want to enjoy themselves and I want the players and the fans to enjoy themselves, that's what life's about.
"I've told the players, I have no fear about a game of football and I want them to have the same philosophy.
"I've learned an awful lot in the last three weeks from what happened to me at my previous club and I hope Leeds have got a better manager on the back of what's happened.
"You want to finish the season strongly. You want to do as well as you possibly can.
"I'm just a bit upset there's only three games left, it's a shame there's not six."
Burnley boss Sean Dyche:
"Leeds showed their experience and that's interesting because the same group of players weren't playing like that two weeks ago because I saw them.
"So that's an interesting one in itself, but we caught them on a night when they were on the front foot and we weren't quite at it.
"There wasn't that freedom in our play tonight, there wasn't that 'front-foot thinking"'.
Live Text Commentary
Full Time The referee ends the match.90:00 +3:16
Free kick awarded for an unfair challenge on Danny Lafferty by Davide Somma. Danny Lafferty restarts play with the free kick.90:00 +2:51 Booking
Booking Jason Shackell goes into the book for unsporting behaviour.90:00 +2:03
Jason Shackell challenges Rodolph Austin unfairly and gives away a free kick. Free kick taken by El-Hadji Diouf.90:00 +0:48
Rodolph Austin concedes a free kick for a foul on Danny Ings. Lee Grant takes the free kick.87:56
Dean Marney crosses the ball, clearance by Jason Pearce.87:24
The ball is delivered by Danny Lafferty, clearance by Jason Pearce.87:16
Kieran Trippier delivers the ball, clearance by Jason Pearce.87:10
The ball is delivered by Junior Stanislas, Jason Pearce manages to make a clearance.85:57
The referee blows for offside. Indirect free kick taken by Kieran Trippier.85:27 Booking
Booking Booking for Luke Varney.85:15
Free kick awarded for an unfair challenge on Kieran Trippier by Luke Varney. Direct free kick taken by Kieran Trippier.84:35
Danny Lafferty delivers the ball, clearance made by Jason Pearce.83:52 Substitution
Substitution Ross McCormack leaves the field to be replaced by Davide Somma.83:21
The ball is delivered by Kieran Trippier.81:47 Booking
Booking The referee cautions Dean Marney for unsporting behaviour.81:26
Dean Marney challenges Ross McCormack unfairly and gives away a free kick. Rodolph Austin takes the free kick.79:35
Free kick awarded for an unfair challenge on Samuel Byram by Kevin Long. Ross McCormack produces a shot on goal direct from the free kick.77:21
Effort from the edge of the box by Danny Ings goes wide of the left-hand upright.75:56
Chris McCann has shot on goal from just outside the penalty box which goes wide of the right-hand upright.
Corner taken left-footed by Stephen Warnock, save by Lee Grant.73:36
The referee blows for offside against Junior Stanislas. Patrick Kenny takes the free kick.71:38
Alex Kacaniklic crosses the ball.70:17
Danny Lafferty crosses the ball, clearance made by Stephen Warnock.69:35
Inswinging corner taken by Michael Tonge, save by Lee Grant.69:23
The ball is sent over by Stephen Warnock, clearance made by Jason Shackell.67:59
Inswinging corner taken by Stephen Warnock.67:13
Dean Marney challenges Luke Varney unfairly and gives away a free kick. Stephen Warnock delivers the ball from the free kick right-footed from left wing, clearance made by Lee Grant.64:57 Substitution
Substitution Danny Ings comes on in place of Ross Wallace.61:53
El-Hadji Diouf provided the assist for the goal.61:53 Goal scored
Goal! - Rodolph Austin - Leeds 1 - 0 Burnley Rodolph Austin grabs a goal from inside the box low into the middle of the goal. Leeds 1-0 Burnley.59:33
The ball is crossed by Samuel Byram, clearance by Kevin Long.58:11
Stephen Warnock sends in a cross, Ross Wallace makes a clearance.54:34
Luke Varney is flagged offside by the assistant referee. Lee Grant takes the free kick.53:53
Centre by Rodolph Austin, Kieran Trippier makes a clearance.52:26 Substitution
Substitution Junior Stanislas joins the action as a substitute, replacing Martin Paterson.49:56
Corner taken right-footed by Michael Tonge.48:55
Unfair challenge on El-Hadji Diouf by Jason Shackell results in a free kick. Ross McCormack produces a cross, Luke Varney takes a shot. Save by Lee Grant.48:40
Kieran Trippier concedes a free kick for a foul on El-Hadji Diouf. Jason Pearce takes the free kick.46:07
Free kick awarded for a foul by Luke Varney on Kieran Trippier. Kieran Trippier takes the direct free kick.45:01
The referee blows his whistle to start the second half.45:00 +3:07 Half time
Half Time The match has reached half-time.45:00 +2:31
Danny Lafferty delivers the ball, save by Patrick Kenny.45:00 +2:00 Substitution
Substitution Danny Lafferty replaces Ben Mee.45:00 +0:48
The ball is sent over by Kieran Trippier, Samuel Byram makes a clearance.43:44
The referee penalises Luke Varney for handball. Lee Grant takes the indirect free kick.43:14
Centre by Samuel Byram.42:31
Effort from inside the area by Martin Paterson misses to the right of the target.41:42
The ball is crossed by El-Hadji Diouf, Lee Grant makes a comfortable save.41:01
Ross Wallace has shot on goal from just outside the area which goes wide of the right-hand upright.40:37
Rodolph Austin gives away a free kick for an unfair challenge on Martin Paterson. Free kick taken by Jason Shackell.39:48
Patrick Kenny takes the free kick.39:48 Booking
Booking Martin Paterson goes into the book.39:34
Free kick awarded for an unfair challenge on Luke Varney by Martin Paterson.38:48
Corner taken left-footed by Ross Wallace, Jason Pearce makes a clearance.38:37
The ball is crossed by Dean Marney, Samuel Byram makes a clearance.35:47
Chris McCann concedes a free kick for a foul on Rodolph Austin. El-Hadji Diouf takes the free kick.35:19
The ball is crossed by Luke Varney.34:21
The ball is swung over by Ross Wallace, clearance made by Paul Green.33:36
Outswinging corner taken by Ross Wallace from the left by-line played to the near post, Michael Tonge makes a clearance.32:23
Stephen Warnock delivers the ball, Headed effort from inside the box by Luke Varney misses to the right of the target.32:12
Rodolph Austin sends in a cross.31:06
Rodolph Austin takes a shot from 35 yards and went wide left of the goal.30:52
Rodolph Austin fouled by Chris McCann, the ref awards a free kick. Rodolph Austin takes the free kick.30:34
Luke Varney takes a shot.29:49
A cross is delivered by Stephen Warnock, Jason Pearce takes a shot. Jason Shackell gets a block in.28:44
Unfair challenge on Ross McCormack by Jason Shackell results in a free kick. Stephen Warnock crosses the ball in from the free kick, clearance made by Jason Shackell.27:35
The ball is sent over by Martin Paterson.26:40
Free kick awarded for an unfair challenge on Dean Marney by Luke Varney. Jason Shackell takes the free kick.24:06
The ball is crossed by Kieran Trippier, clearance by Tom Lees.23:41
A cross is delivered by Stephen Warnock, Chris McCann manages to make a clearance.22:25
Kevin Long fouled by El-Hadji Diouf, the ref awards a free kick. Lee Grant restarts play with the free kick.20:43
Chris McCann gives away a free kick for an unfair challenge on El-Hadji Diouf. Samuel Byram restarts play with the free kick.19:14
Ross McCormack has an effort at goal from outside the box which goes wide right of the goal.19:00
The ball is crossed by Samuel Byram, Chris McCann manages to make a clearance.16:55
Corner taken by Michael Tonge, clearance made by Dean Marney.16:15
The ball is crossed by El-Hadji Diouf, Ben Mee makes a clearance.14:29
Corner taken by Michael Tonge played to the near post.13:53
Corner taken by Stephen Warnock from the right by-line, clearance made by Kevin Long.13:31
Inswinging corner taken by Michael Tonge.12:58
Inswinging corner taken by Stephen Warnock, clearance by Chris McCann.12:44
Ben Mee challenges Rodolph Austin unfairly and gives away a free kick. Stephen Warnock takes the free kick.12:33 Booking
Booking Ben Mee goes into the book.8:32
Ross Wallace crosses the ball from the free kick right-footed from left wing, Tom Lees makes a clearance.8:32 Booking
Booking The referee shows Samuel Byram a yellow card.8:26
Handball by Samuel Byram.6:44
Shot from just outside the box by Chris McCann goes over the crossbar.4:16
A cross is delivered by Rodolph Austin, Ben Mee makes a clearance.2:43
Corner taken by Ross Wallace from the left by-line, Tom Lees manages to make a clearance.2:04
Luke Varney gives away a free kick for an unfair challenge on Jason Shackell. Lee Grant takes the direct free kick.1:26
Luke Varney is flagged offside by the assistant referee. Lee Grant takes the free kick.0:00
The referee gets the game started.
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