Michael Boulding got the rebound after Paul Musselwhite blocked Stuart Campbell's first half free kick.
There was no reply from City so the Mariners ended 1-0 winners in front of a 7,500 crowd watching Neil Mann's testimonial game.
City line-up (4-4-2): Musselwhite; Thelwell, Joseph, Hinds, Delaney; Appleby, Ashbee, Keates, Green; Allsopp, Burgess.
Second half line-up (4-3-3): Fettis; Regan, Whittle, Strong, Peat; Williams, Melton, Mann; Forrester, Webb, Donaldson.