Richie Appleby crossed for Stuart Elliott to head home the only goal of the game just before half time.
Shaun Smith had earlier seen his shot cleared off the line, while Dean Keates and Jon Walters were both denied in the second half.
But a professional 1-0 scoreline was enough for the three points for City who move up to 13th in the table.
City line-up (4-4-2): Fettis; Joseph, Whittle, Anderson, Smith; Appleby (Regan 70), Ashbee, Keates, Elliott; Forrester, Walters.
Un-used subs: Musselwhite, Melton, Dudfield, Webb.