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.
1. Illan Meslier
7 - Distribution wasn't bad, the options were just scarce at times. Made a couple of stops. Good at most set-pieces.
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2. Luke Ayling
6 - Unable to get much going on the right hand side as a wing-back. A couple of fancy bits of footwork.
Picture by Richard Sellers/PA Wire.
3. Liam Cooper
7 - Found himself in a battle, stood up to it well. Couldn't get complete control when the ball was played up top by Burnley.
Photo by Jan Kruger/Getty Images.
4. Pascal Struijk
6 - A similar story to Cooper. A few errors in possession, committed to a challenge he couldn't win to earn yellow.
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5. Diego Llorente
6 - Not as accurate with his passing as he could have been. Burnley tried to target him with physicality but he coped well enough.
Picture by Richard Sellers/PA Wire.
6. Stuart Dallas
6 - Like Ayling, he was kept reasonably quiet in the attacking sense. A few nice passes.
Photo by Richard Sellers/PA Wire.