Some pundits had picked out this fixture as one of the favourites to produce an upset, but goals from Nathan Dyer, Tom Carroll, Modou Barrow, Wayne Routledge and a Bafetimbi Gomis brace ensured the Premier League outfit progressed at the expense of their League Two opponents.
Swansea lost 4-1 at Liverpool on their last trip to Merseyside in the Premier League on Monday, but never looked like suffering another nightmare on the other side of the river, although there was one unwelcome reminder of that defeat for goalkeeper Gerhard Tremmel, whose failed clearance led to Tranmere's first goal courtesy of Max Power.
Dyer's opener after 34 minutes capped a period of total domination, before Tottenham loanee Carroll scored his first goal for the club four minutes after the break.
Barrow - making his first start in a Swansea shirt - added a third just before the hour mark, but Power's fortuitous effort pulled one back in the 70th minute.
Gomis struck 13 minutes from time to leave Tranmere with a mountain to climb, and Routledge and Gomis completed the scoring after Cole Stockton had briefly given Tranmere hope with an 83rd-minute header.
Garry Monk made nine changes from the side that drew 1-1 at QPR on New Year's Day, while Tranmere boss Micky Adams - who had a 13-day stint in charge of Swansea in 1997 - brought Michael Ihiekwe and Stockton into his starting XI.
Despite totally dominating possession for the opening 10 minutes, Swansea had to wait until just after the half-hour mark to break the deadlock through stand-in captain Dyer.
Dyer cut in from the left and found the net at the second time of asking after seeing his first shot blocked in the penalty area, although there were suspicions of handball against the Swansea winger.
It was not until the 36th minute that Tranmere managed to register a shot on target - Power failing to live up to his name as he sent a limp effort straight at Swansea stopper Tremmel from 20 yards.
Having shown glimpses of getting back into the match towards the end of the first half, Tranmere found themselves two down four minutes into the second.
Barrow showed great patience on the right wing and eventually picked out the perfect pass for Carroll, who touched the ball beyond the last defender before tucking under Williams in front of the travelling fans.
Barrow made sure of the win when a Gomis cross left him with a simple tap-in to make it 3-0 after 58 minutes, but Tranmere really should have pulled one back from a goalmouth scramble five minutes later.
Having seen goalkeeping colleague Lukasz Fabianski left red-faced when an attempted clearance rebounded into the net off Adam Lallana at Liverpool on Monday, Tremmel suffered the same humiliation when Power pulled a goal back for the hosts.
Any hope gained by the home side appeared dashed when Gomis sent a low effort beyond Williams from the centre of the penalty area, but Stockton headed a second for Tranmere to lift the noise levels again among the home fans.
However, substitute Routledge slotted home two minutes later to put the result beyond doubt, and Gomis toyed with the Tranmere defence before tucking in a sixth at the death.
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