It needed a smart one-handed save by visiting keeper Joe Lewis to keep out an Adam McGurk header inside the first minute.
McGurk also fired across the face of the goal after a defensive error before Enoch Showunmi header over.
Charlie Lee wasted a good opportunity for the visitors before a deflected George Boyd shot cannoned off the woodwork in the 36th minute. From the subsequent corner Aaron McLean saw a header tipped over by keeper Joe Collister.
Tranmere edged in front in the 45th minute with an excellent low finish from 12 yards by Derby loanee Arnaud Mendy.
James Wesolowski wasted a good chance approaching the hour, Kelvin Langmead grazed the bar with a free-kick and Wesolowski saw a snap-shot well held by Collister as the visitors searched in vain for an equaliser.