Chris Hughton's side took an early lead through Chilean international Jean Beausejour who fired home after a Burke effort struck the crossbar.
Marlon King missed the next best chance of the opening period for the Blues, sidefooting wide with the goal at his mercy from eight yards.
The Clarets equalised four minutes after the break when Bartley collected the ball on the edge of the area and blasted past Boaz Myhill in the Blues goal.
Birmingham pushed for the winner in the remaining minutes and were rewarded with Burke's close-range effort.
The Blues jump from 13th to eighth with the win while Burnley remain precariously above the bottom three in 21st.