Swansea survived a second-half battering to preserve a skin of their teeth victory, which kept their play-off hopes alive.
They took advantage of sloppy defending by Cambridge at a corner kick to score the only goal of the game after only five minutes, and rode their luck as the home team gradually took charge and dominated after the break.
U's boss John Taylor though his team had ended their Abbey Stadium hoodoo when they won their last home game against Bristol Rovers, but the jinx returned with a vengeance.
David Bridges hit the post in the 16th minute, Danny Webb headed against the crossbar in the 83rd and Alex Revell had a shot cleared off the line midway though the second half.
It was a former Cambridge player, Izzy Iriekpen who became the dominant figure in the game.
The big centre back scored the goal in a goalmouth scramble after Cambridge failed to deal with a right wing corner from Roberto Martinez.
Iriekpen then stopped Revell's shot on the line, and in the 78th minute he blocked a goal bound effort Jermaine Easter.
Cambridge, with only three home wins this season and sliding into relegation trouble had most of the possession, especially after the break, but could not make the most of it.
Luke Guttridge did manage to hit the target in stoppage time only to see goalkeeper Brian Murphy pull off a brilliant one-handed save.
A disappointed Taylor said: "We had more than enough chances to have won, we played superbly in the second half, completely dominated, but its fair to say we miss a striker like Dave Kitson, who would have had two or three tonight.
"It was so frustrating. " Swansea boss Brian Flynn praised his side's character as they held on for their first away win for seven months.
"We defended very well under pressure in the second half, " he said.
"Izzy had a tremendous match for us and Brian Murphy in goal, who I brought in for his first match since August, was vital for us at the end.
"We didn't make enough of our first-half possession which is why we are hoping to bring in Paul Connor from Rochdale, before we face top of the table Doncaster on Friday."