Colbeck's right-foot stunner was cruel on Alan Buckley's side after they bossed the first half, but they paid the price for not making the most of the chances that came their way.
The visitors shocked Bradford after just nine minutes when winger Peter Till bravely put them ahead with a close-range finish.
Till was first to react when Ciaran Toner fired over a left-wing cross, but as the Grimsby man forced in the opener under pressure from Luke O'Brien he suffered a nasty-looking leg injury which led to him being stretchered off.
City failed to contain Grimsby's five-man midfield and, with striker Nathan Jarman working hard up front, the visitors should have made more of the early dominance.
Town survived a penalty shout when centre-half Rob Atkinson looked to have handled, but they shot themselves in the foot when full-back Ryan Bennett tapped in Nick Hegarty's goalbound shot and was ruled offside.
In the final minutes of the half, City winger Omar Daley wasted two great openings and the Bantams came out after Stuart McCall's interval pep talk in more determined mood.
Grimsby still looked favourites to take the points until full-back Tom Newey's disastrous mistake let in City's leading scorer Peter Thorne to level after 63 minutes.
With Colbeck breathing down his neck, Newey under-hit a back pass to his keeper and Thorne showed his quality to round Phil Barnes and slide home his right-foot shot from an acute angle.
It was just the lift City needed and with substitute striker Barry Conlon posing more of a threat in the air the game was back in the balance.
City midfielder Kyle Nix had a superb chip tipped away by Barnes and Conlon's mis-hit shot was gratefully gathered by the Town keeper.
With three minutes to go, hard-working Hegarty lifted the ball wide of goal with only keeper Scott Loach to beat and Colbeck made him pay with his late strike.
With City pressing for the winner, Eddie Johnson chested the ball back into the path of Colbeck and his first-time volley from just inside the box fizzed into the bottom corner with Barnes stranded.