Goals by Charles Vernam and former Manchester United youth player Tyrell Warren put the hosts in charge by halftime against the six-time winners of the competition.
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But after thunder and lightning and a torrential rain squall, United finally came to their senses with Bryan Mbeumo's first goal for his new club, offering them an escape route.
Grimsby defended their lead valiantly but Harry Maguire's 89th-minute header sent the tie to penalties.
A nerve-shredding shootout that lasted 18 minutes saw Matheus Cunha have his effort saved when he had the chance to seal it for United. Then Mbeumo saw his effort strike the crossbar to send the home fans into delirium.