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.
7. Adam Forshaw
8 - Ran himself into the ground, played some simple stuff and some nice stuff. Made important interceptions and tackles. Did all that was asked with relish. Photo by Visionhaus/Getty Images.
8. Mateusz Klich
8 - Seemed to grow as the game went on and thrived when it became stretched in the second half. Played clever football and some silky stuff too. Photo by Sebastian Frej/MB Media/Getty Images.
9. Raphinha
7 - Leeds' best attacker in the first half. Less of a threat in the second but he never stopped working. Should have scored in the second half. Photo by Visionhaus/Getty Images.
10. Jack Harrison
8 - Has a player ever needed goal as badly? It was a reward for perseverance and it bred confidence. He played some nice stuff and worked hard. Photo by George Wood/Getty Images.
11. Tyler Roberts
7 - Will know he should have scored at least one but got into good positions, made good runs, helped stretch the defence. Influence waned after the break. Photo by James Williamson - AMA/Getty Images.
12. Joe Gelhardt (sub)
8 - Gave the Burnley defenders a torrid time of it with his runs and physicality. Lovely cross for James' goal, a peach of a delivery. Photo by James Williamson - AMA/Getty Images.