The Mariners, a division below their visitors, bossed the majority of the game, but could not turn a host of decent openings into a goal.
Ross Hannah was denied by Sam Slocombe with only 11 minutes gone as Town did not allow their League Two opponents to settle when in possession.
The industrious Scott Kerr also tested the Iron keeper with a free-kick that swirled through a box full of bodies and, not long before half-time, Lenell John-Lewis wasted a great chance to break the deadlock by firing over the bar after getting in behind the United defence.
Possession was a bit more evenly split in the second half, but still it was Grimsby who dictated most of the play and Slocombe's quick-thinking slid the ball away from the feet of Hannah as he was about to pull the trigger in the 57th minute.
Scunthorpe did not threaten until the final quarter, but substitute Deon Burton was equally as profligate when he could not direct his header on target from Terry Hawkridge's dangerous centre.