Curtis Davies' 12th-minute own goal handed Leicester the lead, but Nikola Zigic finished off a fine move just after the hour mark to get Birmingham back on level terms.
Jeffrey Schlupp headed home a corner 15 minutes from time to seemingly put the hosts on course for victory, but David Nugent was harshly adjudged to have fouled Wes Thomas inside the penalty area in the final minute, with Burke finding the bottom corner with his spot-kick.
The result leaves Leicester winless in nine and still in seventh, a point outside of the play-off places.
The opening goal came following a swift counter attack. Wade Elliott's corner was cleared by Danny Drinkwater and controlled brilliantly by Chris Wood inside the centre circle while under pressure from a couple of defenders.
Wood slipped a pass into the path of Nugent whose 25-yard effort had to be turned behind by Jack Butland.
From the resulting corner, Matty James' cross was headed in by the stooping Davies as he attempted to prevent Wes Morgan from getting to the ball.
Schlupp was presented with a chance to make it 2-0 in the 21st minute when Paul Robinson's header back to Butland dropped short but the England goalkeeper was quickly out to block the shot.
Michael Keane, Andy King and Nugent also wasted chances for the hosts, with Nathan Redmond twice forcing Kasper Schmeichel into action at the other end.
Then, after 61 minutes, Redmond and Paul Caddis carved Leicester open down the right, with the latter providing a low cross which Zigic tapped in from six yards out.
Schlupp was the width of a post away from restoring the host's lead shortly afterwards when his fierce left-foot shot struck the upright.
The Ghanaian did put Leicester back in front after 75 minutes when he rose unmarked to head home James' corner.
It looked to be enough for Pearson's side until the ball bobbled back into the area off Nugent who then tripped substitute Thomas. The referee pointed to the spot and Burke stepped up to level it up once more.