Peter Crouch, also making his first start of the season, added the second with a delicate chip from the final kick of the game at Prenton Park.
Stoke boss Mark Hughes had made seven changes but his side were dominant throughout and should have led by more at half-time.
League One strugglers Tranmere tried to fight back after the break and went close through Jean-Louis Akpa Akpro and Chris Atkinson but the task was beyond them.
Hughes, whose side will now face Birmingham in the next round, said: "I was really pleased with how we went about our business tonight.
"When you have a fixture like this in the early rounds of the cup, if you don't approach it in the right manner and give respect to the opposition, and make sure your work is correct, you can cause yourself problems.
"But from the first minute I could tell the lads were focused on what was required."
Ireland is hoping to rejuvenate his career under Hughes, his former Manchester City boss, after falling out of favour at Villa.
The 27-year-old, who joined the Potters on transfer deadline day, was at the heart of much of Stoke's early attacking play and took his goal well, finishing neatly from a Marko Arnautovic pass.
He was substituted after an hour.
Hughes said: "I was really pleased for Stephen because he is desperate to do well here. I thought he was excellent.
"He got tired and we had to take him off but he got his goal and that was reward for a good time out on the pitch."
Hughes has made a good first impression since taking over at the Britannia Stadium in the summer and his desire to play a more attractive style was again evident, despite the changes in personnel.
Tranmere manager Ronnie Moore said: "We played a Premier League side who have adapted really well in such a short space of time.
"It is fantastic to see how they have adapted, they are a good passing side.
"It was gut-wrenching they scored a second in the last minute but it was a fantastic finish, and that is what they are all about - you give them a couple of chances and they score.
"But overall I was pleased with the way we defended.
"At 1-0 you always have a chance. Our chance came when Chris Atkinson shot wide from a good position.
"We would have deserved it at that stage for hanging in there because we were playing quality opposition - they are level on points with Manchester United."
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