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.
1. Illan Meslier
7 - Beaten by a beauty of a free-kick. Made one save of note but dealt well with corners.
Photo by Stu Forster/Getty Images.
2. Luke Ayling
8 - Brought the ball out well, helped keep Leeds on the front foot in the first half. Dogged in the second half.
Photo by Stu Forster/Getty Images.
3. Diego Llorente
7 - A few nervy moments, harshly dealt with for the free-kick Cornet scored. Battled throughout. Cornet gave him issues.
Photo by Robbie Jay Barratt - AMA/Getty Images
4. Robin Koch
7 - More physical than cultured in his performance but it was an important one from the defender. Fought Wood effectively.
Photo by Stu Forster/Getty Images.
5. Junior Firpo
8 - Better in the first half than the second when he couldn't get forward as much and had less of an influence. First half was excellent.
Photo by Sebastian Frej/MB Media/Getty Images.
6. Stuart Dallas
8 - A performance full of grit and determination but there was good football in there too and a lovely goal. A fine way to mark 250 appearances.