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A Wes Thomas goal, coupled with a fourth consecutive clean sheet, meant the Mariners claimed all three points.
After Saturday's draw with high-flying Exeter, Town passed another test against one of League Two's form teams, taking the lead just after the hour and holding their advantage with a determined display.
The hosts were unchanged for the fourth game in a row, with goalkeeper James McKeown behind a defence of Reece Hall-Johnson, Alex Whitmore, Danny Collins and Luke Hendrie.
Harry Clifton, Mitch Rose and Elliot Embleton again formed a midfield trio, with Charles Vernam and JJ Hooper deployed either side of striker Thomas.
The Grimsby bench was also the same as it had been on Saturday, with John Welsh, Jake Hessenthaler, Jordan Cook, Akin Famewo, Ahkeem Rose, Sam Russell and Ben Pringle all ready for action if required.