The Minstermen went ahead in controversial circumstances on 15 minutes when the ref harshly awarded a penalty against defender Byron Anthony, who was adjudged to have brought Clayton Donaldson down in the box.
The defender received a red card before Bishop scored from the penalty spot, although the ref insisted it was retaken for encroachment into the area by a Rovers defender.
City doubled their lead on 24 minutes when David McGurk headed home Darren Dunning's hanging cross after clever play by Donaldson.
Andy Bishop had a fine shot tipped over by keeper Ryan Clarke, who then did well to save Dunning's stinging volley from 18 yards.
The visitors' main threat came from crosses and from a left-wing free-kick Abdou Sall's header came back off keeper Chris Porter for the tall defender to poke home on 34 minutes.
City put the game beyond the ten-man Rovers with Bishop scoring two goals inside two minutes just before the break.
James Dudgeon's clearance down the middle released Bishop, who outpaced Luke Graham before poking the ball past Clarke from 12 yards as the keeper came out to narrow the angle.
Bishop completed his hat-trick and in doing netted his 20th goal of the season with an exquisite chip over the stranded keeper from an acute angle after latching on to a Donaldson flick.
City wasted possession too often after the break, but scored a fifth goal on the hour when new signing Stephen Thomas arrived at the far post to head home Dunning's pin point cross.
Rovers almost got a second when Zema Abbey charged down Porter's clearance, but defender Dudgeon showed great determination and pace to beat the striker to the loose ball and clear off the line.