The Cherries - who were chasing their 11th consecutive home win - lost goalkeeper Chris Tardif in the 16th minute due to a head injury sustained early in the game. With no backup 'keeper on the bench, midfielder Michael Browning took over.
However, he was rarely troubled as City struggled to even get any shots in. Peter Taylor's side have now gone three league games without a goal.
The home side themselves could have taken the spoils, but were denied by a Paul Musselwhite save in the closing stages.
City line-up (4-4-2): Musselwhite; Regan, Joseph, Anderson, Delaney; Green, Ashbee, Melton, Keates; Jevons (Alexander 65), Elliott.
Un-used subs: Deeney, Whittle, Smith, Webb.