Top scorer Dave Kitson headed his 11th goal of the season to give Cambridge United their second win in four days over the Nationwide Division Three strugglers and it should have sparked a much bigger win.
After the first-half stalemate on a slippery pitch in torrential rain the home side stepped up a couple of gears after the break.
However, despite dominating almost completely as the pace of wingers Shane Tudor and Omer Riza caused the Swansea defence all sorts of problems, they failed to find the finish.
At times they were unlucky particularly when Paul Wanless powered a header against the crossbar in the 47th minute.
There were also good saves from Leigh De Vulgt who cleared off the line from Wanless in the tenth minute and from Roger Freestone who saved superbly from Riza in the 72nd minute.
Generally Cambridge could not find the clinical finish which produced Kitson's goal in the 53rd minute when he steered a header past the keeper from ten yards following a cross from Tudor.
A couple of first-half chances were all that Swansea could manage, and they were also guilty of missing the target.
Jason Smith headed wide in the 17th minute then James Thomas fired across the face of goal on the half hour.
After the game Swansea boss Brian Flynn said: "We couldn't cope with Cambridge's pace after the break and we must learn to defend better under pressure.
"There was an improvement after our 4-0 defeat by Kidderminster but we have a lot of work to do." Cambridge manager John Taylor commented: "We struggled a bit in the first half with a rearranged side after injury.
"It could have still ended up three or 4-0. It was pretty comfortable after we scored.
"Our aim was to be in the top three by this stage of the season but I'll settle for fifth on November the fifth."