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.
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.