Garry Monk's side created chance after chance in the opening 45, but when they did eventually manage to put the ball in the net the goal was chalked off, with defender Michael Morrison adjudged to have fouled debutant goalkeeper Erwin Mulder.
Omar Bogle missed two presentable openings early on, with Lukas Jutkiewicz shooting wide when well placed and defender Harlee Dean forcing Mulder into an impressive stop with a powerful header.
Swansea failed to hit the target with a single effort all night and could and perhaps should have faced the long trip back to suoth Wales with nothing, only for Che Adams to blaze over from a narrow angle with almost 80 minutes gone.
'Swans' boss Graham Potter was under no illusions at full-time as to who had been the better side, telling 'BBC Sport Wales': "I will be the happier of the two managers, that is for sure. Over the course of the game Birmingham deserved to win the match."