Marcelo Bielsa's Whites looked set to suffer a second defeat of the new season after Burnley went ahead in the 61st minute when former Whites striker Chris Wood flicked home a Matthew Lowton shot as Leeds failed to clear a corner.
But United drew level in the 86th minute when Sean Dyche's Clarets survived a ferocious delivery from Raphinha but not the follow-up attempt from Jamie Shackleton which was diverted home by Patrick Bamford with four minutes left.
Leeds are now on two points for the new season, having taken their first point through last weekend's 2-2 draw at home to Everton.
Graham Smyth, the YEP's chief football writer, hands out the scores.
7. Kalvin Phillips
7 - Had to work incredibly hard in the midfield. Silly challenge earned a yellow. Couldn't give Leeds the control they wanted but covered a huge amount of ground. Photo by Jan Kruger/Getty Images.
8. Rodrigo
5 - A sloppy first half. His frustration was evident, tried to get something going in the second half but impact was minimal. Produced a couple of nice passes. Photo by George Wood/Getty Images.
9. Raphinha
7 - Produced the moment that unlocked the Burnley defence. Looked Leeds' best bet for something good. Worked hard all game. Photo by Jan Kruger/Getty Images.
10. Jack Harrison
6 - Lots of effort, not a lot of joy. Unable to link up with Dallas consistently or to great effect. Photo by Stu Forster/Getty Images.
11. Patrick Bamford
7 - Another who got sucked into a battle. He too stood up well. Wasn't at his sharpest but was in the right place to score when needed. Photo by George Wood/Getty Images.
12. Jamie Shackleton (sub)
7 - Plenty of endeavour and got into areas where he could do some damage. Picture by Richard Sellers/PA..