They were rewarded for a bright start at Valley Parade on Tuesday night by going 2-0 in front after 21 minutes, both goals being scored by leading scorer Nahki Wells.
The Bermuda international, who had hit the crossbar with a left-foot shot in the first minute, gave City the lead with a close-range header from Garry Thompson's right-wing cross.
Then, after goalkeeper Jon McLaughlin had prevented an equaliser, coming off his line to block a shot from substitute Ollie Clarke in the 13th minute, Wells put Bradford further in front from the penalty spot after Fabian Broghammer tripped Ricky Ravenhill in the box.
McLaughlin had to come to Bradford's rescue again in the 28th minute, keeping out a point-blank header from Tom Hitchcock after Broghammer's cross found him unmarked.
Bradford added a third in first-half stoppage-time with a looping header from defender Andrew Davies after the visitors failed to deal with Gary Jones' free-kick.
Rovers pulled a goal back in the 50th-minute when Tom Hitchcock took advantage of a mistake by Rory McArdle, but Garry Thompson restored Bradford's three-goal advantage eight minutes later after running on to Wells' through pass.