Blues took the lead in the 12th minute through Matt Green and, although they dominated the game, Leicester had to wait until the 78th minute before equalising with a fine strike by Jamie Vardy.
Boss Nigel Pearson had asked the players to turn the King Power Stadium into a fortress, but their defence was breached in the 12th minute by Birmingham's first attack. Shane Fergsuon crossed the ball from the left and, when Leicester failed to clear, Green pounced to drill home a shot from close range.
Pearson brought on Anthony Knockeart and Lloyd Dyer and switched to a 4-4-2 system.
However, the pattern remained the same with the Foxes totally dominant without breaking through. Knockeart put a shot into the side net from Dyer's low centre and Nugent wasted another chance in the 57th minute with his side-foot shot saved by Darren Randolph.
Foxes goalkeeper Kasper Schmeichel was brought into the action diving to tip a free-kick from Neal Eardley round the post in the 66th minute.
But in the 78th minute the equaliser finally arrived, with Vardy driving a low right-foot shot into the net from the edge of the penalty area after a great run down the left by Dyer.
Four minutes later the Foxes were celebrating again with King producing a clinical finish from Nugent's pass, and a penalty by Nugent in stoppage time after Eardley had brought down Dyer increased Leicester's lead.
Blues hit back with a last-ditch goal from substitute Burke but the Foxes held on for their first home win of the season.