In a goal-less first half Damien Delaney had a header cleared off the line and Marc Joseph shot wide, while Gary Pearson saw a shot saved by Alan Fettis.
The home side opened the scoring 11 minutes after the restart with a goal from Tom Newey.
City were unable to fight back and the Quakers sealed the win in stoppage time with Barry Conlon firing a half-volley past Fettis to make it 2-0.
City line-up (4-4-2): Fettis; Joseph, Whittle, Anderson, Otsemobor (Smith); Regan, Keates (Jevons), Delaney, Melton; Burgess, Elliott.
Un-used subs: Musselwhite, Reeves, Webb.