Caretaker boss Chris Brass saw his side create plenty of chances but a single strike from Jamie Clarke condemned the Shakers to their seventh away loss of the season.
Defender Clarke, formerly of Rochdale, struck five minutes before half-time for his second goal of the season, although it should have been the visitors who led.
Andy Bishop had two good first-half opportunities to put the League Two strugglers ahead. Early in the first-half the striker saw a shot saved by Grimsby keeper Phillip Barnes, before firing wide when he should have done better eight minutes before the break.
In search of only their seventh win of a frustrating season, Bury had an excellent spell of pressure from the half-hour mark.
Nicky Adams saw a powerful shot blocked and moments later Richie Baker fired wide.
Baker also had a shot blocked as Bury pressed but a lapse in concentration saw Clarke give the home side a surprise lead.
Paul Scott headed over the bar and Bishop fired wide on the stroke of half-time but the visitors could find no way through.
Steve Haslam had a shot charged down three minutes into the second-half while Scott wasted another chance when he headed wide to sum up a frustrating evening for Bury.
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