Paul Musselwhite had to be at his best to keep City in front at half time. First, he saved at the feet of Lee Connell after Joseph's mistake put him one-on-one. Then Gareth Seddon's shot took a wicked deflection off Damien Delaney and Musselwhite had to react to tip it over the bar.
City doubled their lead a minute into the second half. Jason Price crossed from the right and Ben Burgess controlled before finishing well.
Stuart Elliott hammered a shot against the underside of the crossbar, while Burgess lobbed over late on, and Bury weren't able to respond to City's two goals.
City line-up (4-3-3): Musselwhite; Hinds, Joseph, Delaney, Dawson; Green (Holt 87), Ashbee, Keates; Price, Burgess, Elliott.
Un-used subs: Harvey, Whittle, Melton, France.