But City were level in the 14th minute through Jason Price, heading in Michael Keane's free kick on his first return to Prenton Park since leaving Tranmere last year.
The winner came midway through the second half when Delroy Facey won a penalty. Keane sent the 'keeper the wrong way to finish it at 2-1.
City now have four points from their opening two games and will be looking to keep up their unbeaten start to the season when Wigan come to Church Road on Wednesday (kick-off 2pm).
City line-up: Duke; Matthews, Hinds, Wiseman, Edge; Price, Fry, Keane (Tomlinson 76), Donaldson (Briggs 87); Facey, Benson.
Un-used subs: Aspden, Hotte, Beattie.