Garry Monk's men threw everything at the visitors in Friday's clash but were unable to find a winning goal against their manager's old club.
Swansea, who failed to record a shot on target, duly missed the chance to leapfrog Middlesbrough to top the Championship table but remain unbeaten.
Birmingham had the ball in the net in the 20th minute, scrambling in with Swansea goalkeeper Erwin Mulder getting the final touch, but the goal was disallowed as captain Michael Morrison had strayed offside.
Mulder saved from Omar Bogle, Jota and Harlee Dean before the break, while Lukas Jutkiewicz failed to make the most of a series of opportunities to grab Birmingham's first win of the season.
And brilliant defending in the closing minutes from Martin Olsson prevented Jutkiewicz from turning in a Jacques Maghoma cross as Swansea held firm for a battling point.