The home side were dominant throughout the first half but could have paid for their wastefulness had it not been for Mullins's winner.
The dribbling of Nathan Redmond and Ravel Morrison caused the Leeds defence problems in the early stages, with the latter having a couple of half-chances and seeing his curling effort rebound off the crossbar.
City missed another good chance midway through the half, when Mullins blazed over after Curtis Davies's header had rebounded to him.
There was still time for another Morrison chance before the break, but the midfielder missed the target after some good skill on the edge of the box.
With 20 minutes to go the home side took the lead, when Mullins was on hand to tap in after Paddy Kenny spilled Redmond's shot.
Chris Burke missed a couple of late chances to double the lead but Birmingham had done enough and won the match by a single goal.