Leeds trailed to Danny Welbeck's 21st-minute strike at Elland Road and looked to be heading for a fourth consecutive defeat which would have kept the Whites in the bottom three with just one game left.
But Pascal Struijk came off the bench to head home a 92nd-minute equaliser after stellar work from Joe Gelhardt and the point moved the Whites out of the drop zone at Burnley's expense.
The Clarets fell to a 1-0 defeat in Sunday lunchtime's clash at Tottenham Hotspur and are now one point behind Leeds but with far better goal difference and an additional game left.
Everton, meanwhile, could have guaranteed their safety in Sunday afternoon's clash against Brentford at Goodison Park but the Toffees fell to a 3-2 defeat as part of a game in which both Jarrad Branthwaite and Salomon Rondon were sent off.
That result has left Frank Lampard's side just one point ahead of Leeds but again with a far better goal difference and an additional fixture remaining.
Everton face Crystal Palace at Goodison Park on Thursday evening before travelling to Arsenal on the final day next Sunday.
Burnley are at Aston Villa on Thursday night and then take on Newcastle United at Turf Moor for their season finale.
Leeds go to Brentford for their final game and these are the new predicted Premier League finishing positions and how the survival chances of each side are rated best on the latest betting markets.
The Clarets fell to a 1-0 defeat in Sunday lunchtime's clash at Tottenham Hotspur and are now one point behind Leeds but with far better goal difference and an additional game left.