The visitors then had a better chance 8 minutes later, but were denied by Paul Musselwhite. A free kick was drilled in, and Paul Tait's header was blocked by Musselwhite from point blank range. Kevin Austin was first to the rebound, but somehow Musselwhite kept his effort out as well in a let-off for the Tigers.
On 25 minutes City took the lead. Rovers keeper Howie saved Anderson's headed from Green's corner, but Gary Alexander scrambled in the rebound to get his 6th of the season.
Elliott had a chance to double the lead but saw his shot blocked, so the sides went in at half time with the score at 1-0.
Dudfield had a glorious opportunity to get what should have been his first goal in over three months early in the second half. Alexander set him free with a through ball, but he knocked the ball wide when one-on-one with Howie.
Musselwhite had to pull off a couple more top drawer saves later in the half to keep hold of the three points, which move City up to 10th in the table.
City line-up (4-3-3): Musselwhite; Regan, Joseph, Anderson, Delaney; Green, Ashbee, Melton; Dudfield (Smith 82), Alexander (Jevons 87), Elliott (Webb 90).
Un-used subs: Deeney, Holt.