Leeds went ahead six minutes before the break as Jack Harrison beat Clarets 'keeper 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.
The win pushed Marcelo Bielsa's Whites eights points clear of the drop zone and Graham Smyth, the YEP's chief football writer, hands out the scores.
13. Dan James (sub)
7 - Took his goal well, worked hard and provided a real threat on the left at times. Should have had an assist but Raphinha missed the gaping goal. Photo by Stu Forster/Getty Images.
14. Paul Tierney (referee)
7 - Wasn't too fussy, didn't buy theatrics, kept a lid on it. Missed one or two things, the free-kick for Cornet's goal was harsh. Photo by Stu Forster/Getty Images.