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, Chris McCann and Martin Paterson all struck to turn the game around, and Steven Davies saw his spot kick saved with the score at 2-1.
Burnley's Charlie Austin had two chances in quick succession in the first period. First City keeper Tom Heaton pulled off a fantastic stop to deny the former Swindon man from Ross Wallace's cross before the former Cardiff goalkeeper kept out another effort shortly after when Austin was found by Vokes' pinpoint delivery.
But then with their first real chance of note in the 35th minute, City took a shock lead.
A fantastic through ball from Davies found the former Barnet man, making his first start under Sean O'Driscoll, and he rounded Grant before rolling the ball into the back of the net to spark delirious celebrations amongst the 247 travelling fans.
Burnley were back on level terms just seven minutes after the break though, Shackell heading a bouncing ball into the bottom corner after a 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.
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's 90th minute volley rounded off the scoring to leave City on course for npower League One.