An inexperienced side was fielded, with Andy Holt the only senior player included before being substituted at half time. Scott Wiseman, Russell Fry, Clayton Donaldson and Nathan Peat also played, while ex-professional Neil Mann was an un-used sub, but the rest were YTS players.
Tranmere forward Paul Robinson - who had scored in each of their last 10 games - hit the post in the 16th minute, with Adam Dickinson putting in the rebound. Paul Brown got another to make it 2-0 at half time.
Clayton Donaldson won a penalty early in the second half, but Russell Howarth saved his strike.
Nathan Hotte headed in Tom Matthews's cross to give City a chance with 10 minutes to go but the home side held on.
City line-up: Pecora, Wiseman, Cattermole, Tomlinson, Byron, Holt (Matthews 45), Russell, Fry, Donaldson, Hotte, Peat.
Un-used subs: Beattie, Mann, Benson.