Garcia Jordi Gomez burst past Franck Queudrue to lash the ball into the net.
Two minutes later Mark Gower could have doubled the lead when Ferrie Bodde sent a 30-yard pass for him to run onto but Maik Taylor raced out to block.
For the opening 15 minutes the visitors scarcely had a touch of the ball as Swansea's passing game continually opened up the Birmingham defence and Gorka Pintado should have added a second but Taylor thrust out an arm to stop his poorly hit shot from going in.
Midway through the half Pintado struck the inside of the post with a header from a Bessone cross.
The Swans suffered a blow in the 33rd minute when Bodde was stretchered off after a collision with Sebastian Larsson.
The enforced change unsettled the Swans and Birmingham equalised in the 41st minute with a downward header by Marcus Bent who met Nicky Hunt's deep cross to score his first goal for the Blues.
In the fourth minute of time added on the Swans regained the lead when Angel Zaragoza Rangel stormed down the right wing. He cut inside and fired low towards goal but the ball took a slight deflection off Radhi Jaidi before being hacked away but the referee had no hesitation in signaling a goal.
Birmingham started the second half with much more energy and purpose and chances began coming their way.
They leveled in the 74th minute when James McFadden crossed and substitute Kevin Phillips headed a simpler equaliser.
Five minutes later he struck his second goal after the Swans' defence failed to clear to turn the game on its head.
Much as the Swans tried to get back on terms they had to settle for their first home defeat of the season.