Tranmere took advantage of Luton's lack of numbers to go into the draw for the fifth round of the FA Cup for the fourth time in five years.
Brian Little's men may be battling against relegation at the wrong end of the Second Division, but they kept up their impressive Cup record when skipper Micky Mellon scored with a shot from the edge of the area in the 81st minute.
Mellon had also got the winner, from the penalty spot, against Chesterfield in the first round, and manager Little was delighted with the Scotsman's efforts.
"That's the second time Micky has got the winner in the Cup for us," he said.
Little was also aware that Luton were heavily hit by injuries and suspensions, particularly in attack, and hardly threatened John Achterberg's goal, with the keeper having only one real save to make when he tipped over Lee Mansell's header in the 31st minute.
"It was a good time for us to play Luton, with all their missing players, and you have to take advantage of the situation," he added.
Mike Newell, his opposite number, agreed, commenting: "The injuries and suspensions were bound to catch up with us. The odds were stacked against us, but we still could have won." Newell was furious that referee Matt Messias turned down a penalty appeal when substitute Keith Keane was pulled down on the edge of the box, but it looked more like a free-kick.
But Luton struggled to create clear chances, and their luck was summed up in the first minute when defender Emmerson Boyce got a shot in on goal following a corner, but saw it deflected wide by a team-mate.
Tranmere were not much better in front of goal, although defender Ian Sharps had a header cleared off the line before half-time. But when Mellon ran on to a loose ball in midfield nine minutes from time, he made no mistake with a thumping shot that flew in off a post to send Rovers through.
"We don't mind who we get in the next round, but one of the big boys at home would be nice for the fans," said Mellon.