Only some indifferent finishing by their leading goalscorer Jason Scotland prevented them producing an upset.
The Trinidadian striker, introduced for the last 30 minutes, carelessly wasted three gilt-edged chances against a vulnerable Birmingham defence.
Birmingham tossed away an ideal opportunity to close the points gap on Championship leaders Wolves with another performance which raised doubts over their promotion creditability.
They are missing crucial players, but the strength in depth of their experienced squad was expected to see them through this difficult patch.
But they huffed and puffed against a Swansea side and suffered the added problem of playing the last 22 minutes with only ten players after David Murphy had been given his marching orders.
Swansea, who had been involved in seven successive draws since being defeated 3-2 by Birmingham City, gave an early indication of why they have shown such form.
They played a lot of attractive football but their finishing was not of the same quality as some impressive approach work.
On the other hand, Birmingham were once again very frustrating as they grinded out another indifferent performance as there appeared little chance of the five-goal thriller from the previous encounter.
Damien Johnson, playing his first home game of the season, was industry personified, but even his non-stop play could not inspire any productive attacks.
Swansea's skill was often very pleasing. Mark Gower often caught the eye, while Ashley Williams added strength to the attack with some surging runs.
Kevin Phillips, who had a very quiet opening 20 minutes, suddenly popped up to remind Swansea of his danger when he flashed a shot just wide following a James McFadden corner.
McFadden then forced Dorus de Vries to make a smart save, while Jordi Gomez produced a dangerous effort which Maik Taylor fumbled away for a corner.
Frustration in both teams then began to boil over with three bookings in the space of two minutes with Stuart Parnaby, Lee Carsley and Gorka Pintado getting yellow cards.
Birmingham's inept finishing was highlighted early in the second half when the off-form Marcus Bent produced a woeful header from a Phillips cross when he found himself completely unmarked just ten yards from goal.
Their task became even harder in the 68th minute when Murphy was sent off after he had been given his second yellow card in the 68th minute for a foul on substitute Scotland.
Scotland then wasted a glorious chance to put Swansea ahead. The talented Leon Britton put him clear but the striker fluffed his left-footed shot to the relief of the Birmingham fans.
The Swansea striker then hesitated when shooting and in the process wasted another opportunity to take him into double figures for the season.