But Bury, whose last league win was back in November, had the chances to have been well in front before they went behind in the 39th minute and substitute Simon Yeo could have had at least a couple within minutes of coming on for Richie Baker in the second half.
But the story of the game was Bury's woeful finishing and some tremendous last ditch goalkeeping by Town's Phil Barnes.
Twice he faced up to a Bury player when all seemed lost, but he denied first Baker by blocking with his legs and then did similarly to deny the tricky Nicky Adams in first-half added time
By this point however, Grimsby were in front with a goal totally out of character with the rest of the match.
Apart from a shot from midfielder James Hunt which forced a save from Bury keeper Jim Provett, there had been nothing to trouble the visitors' keeper.
But when Clarke found himself some space 25 yards out, he let fly with a scorching drive which flew into the net off the inside of the far post.
Nobody was more surprised than the supporters in a thin crowd, who had witnessed nothing up to this point.
But just when you thought that this might have lifted the Mariners, they were unable to create a single chance in the second half and it was the visitors who went closest.
Yeo, who had troubled Grimsby in the past, squandered two great chances and then even in added time at the end of the game, Bury could have gained a point but Adams looped his header wide.
This result lifts Grimsby into the top half and they are eight games unbeaten without ever reaching anything like the level of which they are capable of, while Bury now out of the FA Cup have a fight on their hands to maintain their Football League status.