Scottish striker Ross McCormack, making only his fifth start of the season, provided an emphatic first-half finish as Leeds clinched their first win in six matches to climb seventh in the table, level on points with Nottingham Forest, who play Scunthorpe later in the day.
Simon Grayson's side leapfrogged Burnley in the process and inflicted on the Clarets their first defeat in five matches, leaving them two points adrift of the last play-off spot that Forest can all but make their own with victory over Scunthorpe.
Burnley had won three of their last four games to give their travelling fans genuine hope of gatecrashing the top six and played their full part, with only a last-minute fingertip save from Leeds goalkeeper Kasper Schmeichel from substitute Ross Wallace's free-kick denying Eddie Howe's side a share of the points at Elland Road.
Midfielder Neil Kilkenny, on-loan Sunderland left-back George McCartney and striker McCormack returned for Leeds, while Burnley were unchanged.
Leeds settled quickly and Burnley goalkeeper Brian Jensen was made to stretch to hold McCormack's delicate chip in the sixth minute, while Paynter fired inches wide soon after.
Burnley midfielder Wade Elliott's shot was blocked by Leeds skipper Richard Naylor at the other end following Chris Eagles' marauding run.
Burnley were denied strong claims for a penalty in the 18th minute when Leeds defender Andy O'Brien climbed over the back of Chris Iwelumo and Kasper Schmeichel was then at his best to smother Burnley skipper Michael Duff's angled shot.
Eagles' long-range effort flew over the crossbar after another incisive run from the Burnley midfielder and Max Gradel's powerful drive kept Jensen on his toes as the action flowed.
McCormack curled a 26th-minute free-kick just wide and from his cross moments later Billy Paynter's fine header was turned away at full stretch by Jensen.
Leeds then made the breakthrough. Bradley Johnson's raking first-time pass caught Burnley's back four flat-footed and McCormack, played onside by Burnley centre-half Andre Bikey, raced through to shoot low inside Jensen's bottom right-hand corner in the 33rd minute.
Kent referee Phil Crossley booked Clarets skipper Duff for pulling back McCormack as Leeds held on to their lead at the break.
Paynter headed straight at Jensen from McCormack's cross in the 54th minute and then shot into the side netting after rounding the goalkeeper.
Chris McCann's high tackle on Gradel sparked a mass melee in the 57th minute for which the Clarets midfielder was booked.
O'Brien's header from McCormack's subsequent free-kick was brilliantly saved by Jensen, who then matched that with a full-length dive to keep out the Scottish striker's deflected shot as Leeds chased a second goal.
But Burnley were playing their part and midfielder Jack Cork thundered a 25-yard effort inches wide.
McCormack had the chance to wrap the points up in the 83rd minute when sent clear by Gradel, but he dragged his shot wide, while Burnley thought substitute Ross Wallace had rescued a point when Schmeichel tipped over his last-minute free-kick.
Home substitute Sanchez Watt spurned two late chances to score the second killer goal, failing to trouble Jensen when well placed in the area, but Leeds ran out winners to just about keep their season alive when they head to champions-elect QPR on the final day.