Leeds went ahead six minutes before the break as Jack Harrison beat Wayne Hennessey at the second attempt after being gifted possession by James Tarkowski.
Burnley drew level nine minutes into the second half via a fine free-kick from Maxwell Cornet, awarded after a dubious foul by Diego Llorente some 25 yards out.
But Leeds went back in front with 13 minutes left when Stuart Dallas curled home a beautiful finish from the edge of the box on his 250th outing for the club and two substitutes combined for United to bag a third in the 93rd minute as Dan James headed home a Joe Gelhardt cross.
After a frustrating start, Leeds threatened for the first time in the 15th minute when Stuart Dallas cut in from the left and unleashed a low shot that keeper Hennessey eventually gathered.
Two minutes later, Raphinha attempted to lob Hennessey from the halfway line and his brilliant attempt clipped the top of the bar before bouncing behind.
Leeds began to turn the screw and a superb Raphinha cross was flicked goalwards by Junior Firpo only for Hennessey to turn the ball behind via an excellent save at his near post.
After a brief Burnley rally, Raphinha cut inside before firing over from the edge of the box but at the other end only a five save from Illan Meslier kept out Chris Wood's attempted finish to a Charlie Taylor cross.
Back came Leeds and the Whites squandered a glorious chance in the 34th minute when a lovely cross from Luke Ayling picked out Tyler Roberts who sent a free header back across goal but wide.
But United went ahead in the 39th minute when Tarkowski's poor pass back let in Harrison who saw his first shot saved by Hennessey but the winger smartly converted the rebound with his right foot before the Clarets keeper had time to recover.
Leeds quickly pressed for a second and a Roberts shot was blocked before a lovely move ended with Firpo blasting a good opening over the bar.
Another Clarets error then let in Roberts whose shot from a wide angle was straight at Hennessey.
But Burnley were inches away from equalising on the strike of half-time when Taylor stormed down the left and sent in another fine low cross which Wood hammered just over the bar looking for a first time finish.
Returning Clarets star Maxwel Cornet was brought on during the interval to replace Johann Gudmondsson but Leeds squandered a chance to double their lead in the 52nd minute when Mateusz Klich played in Roberts who lost his footing when and eventually fired over after looking to recover.
Two minutes later, Burnley were level after being awarded a soft free kick, Llorente judged to have pulled down Cornet following a tangle 25 yards out.
Cornet stepped up to take the free-kick and sent a brilliant effort into the bottom left corner.
Leeds then lost Roberts to injury a few minutes later but Elland Road erupted as Gelhardt was brought on as his replacement. James followed him on for Harrison shortly after.
Ten minutes later, a James cross from the left whizzed through the area to Raphinha but the Brazilian side footed his shot into the side netting from a tight angle.
Burnley threatened shortly afterwards and Cornet just failed to slide home Wood's cross from the right at the far post.
And with 18 minutes left United went back in front as Raphinha and Klich worked a short corner to Dallas who produced a brilliant curling shot from the edge of the box that Hennessey could only watch nestle into the left hand corner of the net.
Leeds continued to press in search of a third goal as Gelhardt caused problems for the Clarets defence and a superb cross from Gelhardt lead to United bagging a third goal in the 93rd minute.
Terrific work from Klich set Gelhardt away down the right and the teen delivered a brilliant cross that was met by the head of James whose attempt looped into the right hand corner of the net to send Elland Road wild.
Leeds United: Meslier, Ayling, Llorente, Koch, Firpo, Forshaw, Dallas, Raphinha, Harrison (James 62), Klich, Roberts (Gelhardt 58). Subs not used: Klaesson, Hjelde, Moore, Jenkins, Bate, Summerville, Greenwood.
Burnley: Hennessey, Lowton, Tarkowski, Mee, Taylor, Gudmundsson (Cornet 46), Westwood, Cork, McNeil, Lennon (Rodriguez 80), Wood (Vydra 84). Subs not used: Norris, Stephens, Collins, Pieters, Bardsley, Long.
Referee: Paul Tierney.
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Thank you Laura Collins