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.
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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