A trademark long-range shot from Gary McAllister and two quickfire goals from the Moroccan forward Youssef Chippo just before the interval gave the Sky Blues a 3-1 half-time lead, Scott Taylor replying for the First Division strugglers.
But, despite some intense pressure, the Premiership outfit were unable to make further inroads into Tranmere's lead and went out 6-4 on aggregate.
The Tranmere manager, John Aldridge, said: "For 43 minutes I thought we were okay and two minutes of madness really put us under the cosh. Half-time couldn't come quick enough for me really. It gave us time to regroup. We were panicking at the time and as a manager I was as well.
"At half-time it did look ominous. We had a talk after I'd calmed myself down. Then they went out and showed an awful amount of character and the players deserve a lot of credit."
Derby, held to a 0-0 draw at Third Division Swansea in the first leg, recovered to win 3-1 at Pride Park. Esteban Fuertes opened the scoring after 26 minutes, the substitute Dean Sturridge (66) and Vas Borbokis (76) were all on target to help ease the memory of Saturday's 5-0 surrender to Sunderland.
Derby's manager, Jim Smith, made five changes from the side humiliated by Peter Reid's men and all Swansea could manage was a late consolation by their captain Matthew Bound from the penalty spot.
The Leicester City youngster Stefan Oakes enjoyed a night to remember at Filbert Street as the first two goals of his career helped them overcome First Division Crystal Palace 4-2 and seal a 7-5 aggregate win.
Oakes could have had a hat-trick but he missed a second-half penalty as the Leicester fans found time to continue to call for the head of chief executive Barrie Pierpoint. Ian Marshall (53) and Graham Fenton (79) scored in between Oakes' double - a 19th-minute penalty and another two minutes from time - with Steve Thomson (61) and Lee Bradbury (86) replying for Palace.
Leicester's manager Martin O'Neill revealed afterwards that he had not written programme notes before the game because he feared they would be censored because of the current boardroom feud.
Bradford twice came from a goal behind at managerless Reading to go through on away goals after being taken to extra time by the Second Division side. Keith Scott and Barry Hunter scored for Reading with Dean Saunders and David Wetherall replying.
Blackburn's 3-1 victory over First Division Portsmouth at Ewood Park completed a 6-1 aggregate verdict. Alan McLoughlin gave Pompey hope with a seventh-minute opener but Rovers responded with goals from Damien Duff (59), David Dunn (64) and Kevin Gallacher in injury time.
Nottingham Forest earned a 0-0 draw at Bristol City to go through 2-1 on aggregate.