The Minstermen started the game on top and were unlucky not to take the lead early on, with Russell Penn seeing his stinging drive blocked by Steve Mildenhall from point-blank range.
Rovers had struggled in the opening exchanges but they had a goal chalked off prior to the break with John-Joe O'Toole was adjudged to be in an offside position at the far post.
The visitors then wasted a big chance five minutes before the interval when Matt Harrold nearly capitalised on a moment of hesitation from Nick Pope but he could only head Tom Lockyer's cross against the far post.
With 25 minutes left to play City almost took the lead, with Mildenhall at full stretch to tip Keith Lowe's header over the crossbar.
The visitors threatened for a winner in the closing stages but substitute Chris Beardsley failed to direct Lockyer's cross on target.