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The Mariners were forced to play almost the entire game a man short after Luke Waterfall was shown the red card inside the opening 10 minutes.
But after withstanding long spells of home pressure in the first half, Town were the better side after the break and got the vital breakthrough when home defender Mal Benning volleyed into his own net.
One change was made to the Grimsby side after the New Year's Day victory over Salford, with Moses Ogbu replacing the injured Jordan Cook.
Mansfield: Logan, White, Benning, Preston, Pewwarce (C), Bishop, MacDonald, Mellis, Hamilton, Maynard, Rose
Subs: Olejnik, Clarke, Khan, Smith, Cook, Knowles,
Town: McKeown (c), Hendrie, Davis, Waterfall, Gibson, Hessenthaler, Clifton, Robson, Ogbu, Vernam, Wright. Subs: Russell, Öhman, Pollock, Hewitt, Whitehouse, Rose, Cardwell.
Referee: James Oldham
Attendance: 5,840 (1337)