The Potters, who made seven changes for the tie, will now travel to Birmingham in the next round.
Hughes 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.
"There was a good tempo to our passing and our play. We played some really good stuff at times.
"We were in total control of the game, which was really pleasing."
Ireland is hoping to rejuvenate his career under his former Manchester City manager Hughes after joining on a season-long loan from Aston Villa.
The Irishman was at the heart of much of Stoke's good attacking play in the first half, in which they squandered numerous opportunities to make the win more comfortable, before being removed 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."
Despite Stoke's superiority, their inability to take chances in the first half gave Tranmere encouragement after the break.
Jean-Louis Akpa Akpro and Chris Atkinson both had good opportunities but the Merseysiders, who are 19th in League One, were unable to get back into the game.
Manager Ronnie Moore said: "It was gut-wrenching they scored a second in the last minute but it was a fantastic finish, and that it was 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."
- Sports & Recreation
- Peter Crouch
- Stephen Ireland