Martin Paterson scored an 85th-minute equaliser to prevent Peterborough moving off the foot of the table at London Road.
Charlie Austin scored after just 64 seconds on his return from injury but Tommy Rowe's two second-half strikes appeared to have secured a vital victory for Posh.
But substitute Paterson pounced late on to secure a point for the Clarets and leave Darren Ferguson's side bottom of the Championship.
Austin had been missing since the end of December due to a hamstring injury and was joined in the starting XI by Michael Duff, who had also returned from a hamstring problem.
But Ferguson's plans went out the window in just the second minute as Austin fired in his 24th goal of the campaign, volleying in Ross Wallace's cross from the left.
Peterborough recovered well from the setback and came close to an equaliser in the ninth minute as Michael Bostwick headed over from Grant McCann's free-kick from the right.
Austin got his second sight of goal in the 11th minute but his snapshot from the edge of the penalty area was straight at Posh goalkeeper Robert Olejnik.
After a relatively quiet opening period, the chances came thick and fast at the start of the second.
Within three minutes the dangerous Lee Tomlin fired in a curling effort from the edge of the area that narrowly missed Lee Grant's left-hand post.
Dwight Gayle then forced Grant into a save down to his right two minutes later, with the keeper holding well.
A slip in the Clarets' defence from Danny Lafferty gave Gayle another opportunity for Posh but he could only fire wide from the edge of the area.
And Posh equalised with 25 minutes left as Burnley failed to clear a corner and Rowe fired in from six yards.
Four minutes later Rowe ran 60 yards with the ball and found Gayle whose shot was diverted back into Rowe's path and he drilled in his second from inside the area.
Paterson levelled for Sean Dyche's side with five minutes left as he sent Chris McCann's cross past Olejnik from close range.
He nearly added a dramatic winner in the dying stages but Olejnik saved brilliantly from his close-range header.
