Micky Mellon`s side had performed extremely well to hold SkyBet League One side Peterborough United to a 1-1 draw at their London Road home on November 4th, but a dreadful opening 25 minutes for the hosts killed the tie off as a contest and it was then down to damage limitation.
The visitors came out looking in determined mood and took the lead after 16 minutes when they capitalised on a poor back pass to open the scoring through Danny Lloyd.
The former Stockport County striker made it 2-0 just 6 minutes later from close range after a neat cushioned header across goal from ex-Newcastle United star Steven Taylor.
Things went from bad to worse for Tranmere 3 minutes later when they had midfielder Jay Harris shown a straight red card for a reckless challenge on Marcus Maddison.
However, Mellon`s men managed to stem the tide and actually begin to threaten Posh at the other end on occasions before the half-time break.
The second half saw Rovers battled hard and had a good deal of possession, but fell three behind after 67 minutes when a free-kick from Maddison was headed in unchallenged in the six-yard box by Posh skipper Jack Baldwin.
That effectively killed the tie off as a contest and 6 minutes later it was 4-0 when another set piece from Maddison was parried by Rovers` keeper Scott Davies but he could do nothing about Lloyd following up to complete his hat-trick from close range.
Two minutes later, Peterborough made it a nap hand when Davies brought down sub Jermaine Anderson to concede a penalty which was subsequently converted by Maddison.
Visiting keeper Conor O`Malley was called into action with 10 minutes remaining to save from Andy Cook and then James Norwood as Tranmere, to their credit, still pushed forward looking to reward their fans, who stayed with them throughout, with a consolation goal.
The victory means Peterborough will play Tranmere`s league rivals Woking, who won 3-0 at Bury in their replay in the Second Round.
The match will take place at The Laithwaite Community Stadium on Sunday, 3rd December, with a 2pm kick-off, with the BBC selecting the game for coverage on the `Final Score` show.