Defeat for the Shakers must have been hard to swallow after they led through Danny Reet's 13th-minute strike, and dominated proceedings, only to be denied by a string of top-class saves from Town keeper Steve Mildenhall.
But patience and perseverance paid off for the visitors as they kept the pressure on leaders Wycombe.
The early pressure came from Bury. Sought-after midfielder Simon Whaley was dangerous on the right wing and nearly supplied the first goal in the second minute, only for Mildenhall to collect his wicked cross and avert any danger.
The home side continued to press and after Grimsby had failed to clear a Whaley corner from the left, fellow midfielder David Flitcroft forced Mildenhall to make an excellent fingertip save from his sweet right-foot strike.
The opening exchanges were eventful and Grimsby striker Michael Reddy twice had penalty appeals turned down, before Reet went agonisingly close to giving Chris Casper's men the lead.
A clever ball from David Buchanan beat the offside trap but, despite wrong-footing the last defender, Reet hit the foot of the left-hand post and the visitors were spared.
However, it wasn't long before Bury were in front. Whaley sent in a corner from the right, skipper Dave Challinor rose highest at the back post and headed into the path of Reet who, from close range, only needed to hit the target. He did it with ease.
The advantage was nearly doubled a minute later. Challinor headed over inside the six-yard box under pressure, before Grimsby hit back against the run of play.
A hopeful free-kick, taken close to the halfway line, somehow evaded Bury's defensive line and Gary Jones was able to head past Neil Edwards, unmarked at the back post.
Whaley fired over in first-half injury-time, doing well to get a shot away on the turn while under pressure, but honours were even at the interval.
The second-half began with a streaker holding up the game for two minutes, before Whaley rode a crunching challenge to set Reet free. But the on-loan Sheffield Wednesday hitman saw Mildenhall produce an excellent block with his legs to deny him.
A minute later Nicky Adams whipped in a testing cross from the right and Reet, sliding in at the near post, just failed to hit the target.
Just after the hour mark Reet saw a shot blocked as he continued to search for his second goal, before centre-back Paul Scott met a Whaley corner sweetly, but his header was well-saved by Mildenhall on the line.
At the other end Tom Newey saw a curling free-kick saved by Edwards, who got down well to collect the ball at the near post.
Whaley crossed from the right in the 65th minute, Reet dummied in the six-yard-box, but Tom Youngs, outstretched and sliding in, failed to get the slightest of touches which would have surely given the Shakers the lead.
But there was heartache for the home side after they failed to clear their lines. A free-kick into the box saw Edwards fail to collect the ball and the Bury defence took too long to get rid of the danger. Cohen gratefully fired home, unmarked, when the ball rolled out to him in the penalty area.
Bury pressured in the final stages, but their best chance, a header from Scott, was saved by Mildenhall.