City had plenty of opportunities to secure all three points, but poor finishing ensured they extended their recent run to just one win in their last five matches.
There were two changes in the Swansea line-up with Willy Gueret regaining his place in goal and Kevin Amankwaah in for Dennis Lawrence, who was on international duty with Trinidad & Tobago.
The visitors made three changes from the team that started the 2-0 defeat at Bradford City, with Chris McCready, Paul McLaren and Steve Davies coming in.
Tranmere started brightly and Chris Greenacre had a shooting chance as early as the third minute, but Gueret was swiftly off his line to block.
The Swans hit back two minutes later with a Leon Britton cross that Rory Fallon headed down towards the bottom corner, but Rovers goalkeeper Gavin Ward did well to push it around the post.
Leon Knight had a gilt-edged chance to open the scoring but he could only glance a Tom Williams cross wide from a good position.
Swansea's frailty at the back was shown when an unmarked Shane Sherriff headed a Chris Shuker corner against the woodwork without any challenge from a home defender.
A superb tackle by Robbie Stockdale on Andy Robinson prevented a certain goal as the first half drew to a close.
Five minutes into the second period a free-kick from McLaren went through to Greenacre but the little striker stubbed his shot and Gueret gathered.
A superb Swansea build-up in the 55th minute should have finished with a goal but when the cross came in, Robinson blasted it high over the crossbar from point-blank range.
Robinson squandered another chance in the 79th minute after a run by Lee Trundle, who had replaced the ineffective Knight, but Rovers comfortably held out for the point that lifted them to second place in the division.
Swansea manager Kenny Jackett is still under pressure but this time it was the attack that let him down, as the defence, with Alan Tate at centre-back, seemed far more solid than in recent matches.