The visitors looked on course for a vital three points when Albert Adomah became the first City player to net on the road since New Year's Day.
But Jason Shackell's header, Chris McCann's first goal since he scored a late winner when the sides met back in October and a late strike from Martin Paterson earned Burnley only their second win in 13 games.
It took the home side's attacking duo just two minutes to create the first chance of the match between two out-of-sorts sides with former Bristol City loanee Sam Vokes volleying wide after a lovely flick from top scorer Charlie Austin.
The Robins defence were struggling to cope with Austin and he fired another shot across goal as his side searched for just their second home win in 2013.
But then with their first real chance of note in the 35th minute, City took a shock lead through Adomah.
A fantastic through ball from Steve Davies found the former Barnet man, making his first start under Sean O'Driscoll, and he rounded Lee Grant before rolling the ball into the back of the net to spark delirious celebrations amongst the 247 travelling fans.
They were back on level terms just seven minutes after the break though, Shackell heading a bouncing ball into the bottom corner after a Junior Stanislas corner evaded the Robins defence.
But the home side's nine minute turnaround was complete in the 61st minute when McCann rounded off a passing move with a drilled left-foot finish into the bottom corner. And the Clarets continued to push for a third with Austin's curling effort landing on top of Tom Heaton's net.
With their survival prospects looking to be in tatters, O'Driscoll introduced Neil Kilkenny as they searched for their second late equaliser this week.
Joseph Mills, back in the side for the first time since December, then handed the visitors a chance to do just that when he fouled Adomah inside the box.
Grant though guessed the right way and made a fantastic save to his right to keep out Davies' penalty.
And Paterson completed the scoring when he volleyed home in the 90th minute to leave City on course for League One.