Blues, who began the night with three games in hand over most of the teams around them, had led through Jean Beausejour's second-minute strike and would have been winning more comfortably had Marlon King not been guilty of an alarming miss.
Blues looked like being made to pay for their profligacy again as Marvin Bartley's strike early in the second half put Burnley level only for Burke to pop up with an injury-time winner.
With less than two minutes gone, Blues were ahead. A slick passing move saw striker Nikola Zigic flick the ball through for Burke in a delightful one-two and when Burnley goalkeeper Lee Grant's touch deflected his shot onto the bar, Chile winger Beausejour reacted quickly to slam home the rebound from the left-hand side of the area.
Zigic should have done better than head Beausejour's 13th-minute corner wide when well-positioned as the Blues kept up the pressure.
Keith Treacy's header moments later was not far off target at the other end as Burnley managed a rare foray forward.
Zigic again came close with a header when he met David Murphy's 25th-minute free-kick but directed the ball just wide.
It would have been 2-0 in the 35th minute had King not produced one of the misses of the season so far.
Zigic showed alertness and tenacity to rob David Edgar of the ball and charge forward and when the forward selflessly squared the ball to his strike partner, King somehow sidefooted wide from around eight yards with keeper Grant stranded.
Blues were made to pay for not taking their chances as Burnley levelled less than four minutes into the second half. Careless play led to the Clarets winning a corner and when Ross Wallace's ball was not properly cleared, midfielder Bartley steadied himself before rifling in a shot from 20 yards which was deflected beyond Boaz Myhill's reach.
Murphy's cross from the left forced Grant to swat the ball away frantically as Blues looked to reassert themselves, with Keith Fahey's first-time volley not far off target minutes later.
Fahey again went close as the home support grew impatient while Murphy was booked for a late challenge on Kieran Trippier.
Zigic sent a header looping just over from Stephen Carr's deep cross but Burnley always threatened to land a sucker-punch.
King saw a tame effort saved either side of speculative shots by Burnley pair Jay Rodriguez and Treacy.
Zigic was then guilty of a glaring miss of his own four minutes from time, heading substitute Nathan Redmond's superb cross down and out rather than into the bottom corner.
Some home fans had started filing out of the stadium when Burke hit the winner a minute into added time. Redmond placed the ball perfectly into the path of Murphy and when his teasing low ball left the defence flat-footed, Burke slammed the ball home to secure a dramatic three points.