Steve Jennings turned an Andy Robinson shot into his own net in the 81st minute, although Robinson said later that he would be claiming the goal.
It was a disappointing match between two promotion-chasing sides.
Tranmere certainly had the better of the chances but were wasteful in front of goal and also found City keeper Dorus de Vries in sound form.
Vries made excellent first-half stops from Antony Kay and Gareth Taylor.
It took Swansea 40 minutes to test home keeper Danny Coyne via a 30-yarder from Marcos Painter.
Chris Greenacre blazed over from close-range after an hour, while at the other end Jason Scotland mis-hit his shot when set up by Robinson.
Calvin Zola, who went on as a 69th minute substitute for Tranmere, headed wide in the 72nd minute and then a Greenacre effort from a Paul McLaren free-kick was deflected away for a corner.
Tranmere's bad luck continued as Ian Goodison hooked the ball over from close range in the 77th minute.
The own goal lifted Swansea and they almost doubled their advantage in the 85th minute when Coyne did well to turn an effort by substitute Guillem Mayol Bussy over the bar.
Tranmere finished strongly but a Shane Sherriff header following a free-kick was turned aside by de Vries, before Sherriff fired inches wide of the target from 25 yards.