City had the upper hand in the second half however, hitting the woodwork three times before getting the equaliser 20 minutes from time, with youngster Steve Burton heading in a corner.
Gary Bradshaw then headed home Ben Morley's cross for the winner.
City are two points behind leaders Macclesfield, each with a game left to play. If City win at bottom-of-the-league Halifax and the Silkmen fail to beat mid-table Hartlepool, the Avon Insurance Division 3 trophy will be coming to Boothferry Park.
City line-up: Glennon; Morley, Peat, Burton, Holt; Matthews, Kerr, Whitmore, Beresford; Rowe, Bradshaw.
Subs: Bowsley, Tomlinson, Turnbull, Kaveney, Crutwell.