It was a fifth away success for City to boost hopes of a play-off spot but they rode their luck somewhat in the second half in the face on incessant Tranmere pressure.
The goal came in the tenth minute and was a14th of the season for Lita. He neatly turned defender Ian Goodison after a throughball from Luke Wilkshire and planted an angled left-foot shot out of the reach of Russell Howarth.
Tranmere gradually improved their game as they went in search of an equaliser.
City keeper Steve Phillips produced a superb blocking save from a Ryan Taylor free-kick in the 14th minute and did even better in the 24th minute when he kept out a close-range Iain Hume effort after good work by Paul Hall.
Rovers laid siege to the City goal for much of the second period but came up against a defiant rearguard.
But it needed a goal line clearance by Mickey Bell from an Ian Sharps header in the 59th minute to keep them ahead.
City could easily have doubled their lead on 66 minutes when a Bradley Orr shot was deflected onto the top of the net. Then, in the 73rd minute, home keeper Howarth had to be at his best to deny Wilkshire.
Tranmere had a penalty appeal turned down for an apparent handball in the 79th minute and the frustration grew as they failed to break through.
Hall was outstanding for the Merseysiders while Stephen Brooker deserved to finish on the winning side after working tirelessly for the visitors.