John Anderson had a shot cleared off the line, but the equaliser came just after the half-hour mark. Jon Walters held off a defender's challenge to turn and fire the ball into the net and restore parity.
Three minutes later City went in front. Walters' cross was played back across by Carl Regan, and Stuart Elliott dived in at the back post to head City into the lead.
A matter of seconds had passed after half time before City added to their lead. Dean Keates passed to Regan on the right wing, and his cross again found Elliott to head home his 11th of the season.
Looking for his hat-trick, Elliott had a long range shot saved by Moss, and then Ben Burgess went just wide from the parry.
Alan Fettis saw off Bournemouth's chances of a comeback, making a great save from substitute Brian Stock's header.
15,816 watched at the KC Stadium as City won 3-1 - the third victory from the last four games.
City line-up (4-4-2): Fettis; Joseph, Whittle, Anderson, Smith (Otsemobor 64); Regan, Keates (Reeves 87), Delaney, Elliott; Walters (Melton 77), Burgess.
Un-used subs: Musselwhite, Jevons.