The League Two side may have fancied their chances of an upset after seeing Swans boss Garry Monk make nine changes from his side’s 1-1 draw with QPR in the Premier League.
However, with 80 places between the two sides, Swansea showed the superior class as Nathan Dyer, Tom Carroll and Modou Barrow fired them into a 3-0 lead.
Max Power pulled one back for the hosts after an error from goalkeeper Gerhard Tremel but Swansea put the result beyond doubt with just under 15 minutes remaining as Bafetimbi Gomis finished off a swift counter-attack from the visitors.
Cole Stockton pulled one back for the hosts but Wayne Routledge and Gomis rounded things of the visitors as they booked their place in Monday’s fourth-round draw.
The visitors looked to stamp their Premier League class on the game from the very first whistle as Gomis and Marvin Emnes both had early chances, but both efforts failed to test Tranmere keeper Fon Williams.
Gomis was proving a real handful for the home defence and he would have broken the deadlock after Modou Barrow’s cross found him in space in the box but his left-foot shot was tipped over superbly by Williams.
Swansea were dominating proceedings and created yet another opening and this time Gomis turned provider, crossing for Jay Fultonn at the far post but the midfielder could not get a shot off on goal.
Monk would have been wondering whether his side would be made to pay for their wastefulness in front of goal after Emnes broke through the Tranmere defence before shooting inches wide of the right-hand post.
However, the visitors were not to be denied any longer as Swans captain for the day Nathan Dyer fired past Williams with a right-footed strike from inside the penalty area after seeing his first shot blocked.
Having not seen much of Tranmere as an attacking force, Micky Adams’ side responded well to going a goal down and created chances for Danny Holmes, Power and Steve Jennings but it was Swansea who went into the break with a deserved lead.
Despite their decent finish to the first 45 minutes, there was to be no let up for Tranmere as Swansea doubled their advantage minutes after the interval after good work from the impressive Barrow, found Tom Carroll, who fired past Williams from close range.
Things were about to get worse for Tranmere as Swansea showed their class and it was Barrow’s turn to get on the scoresheet as he tapped home Gomis’ cross from close range to put his side three goals to the good.
Tranmere managed to pull one back after Tremmel’s clearance cannoned off Power and looped into the net but any hope was soon distinguished as Gomis finished off a lightning counter-attack to score his side’s fourth of the afternoon.
The hosts picked up another after Stockton latched onto Marcus Holness’ headed knockdown to tap home from close range, but it was Swansea who had the final word as Routledge’s right-footed shot found the bottom corner and Gomis picked up his second of the match after good work from Ashley Richards.