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A very young Mariners side produced a superb performance against their more experienced opponents and Montel Gibson's penalty put the hosts in front nine minutes before the break.
It was a night to remember for 15-year-old Louis Boyd as he came off the bench to become Town's youngest-ever first-team player before also becoming the club's youngest goalscorer as he made it 2-0.
But the visitors struck back through Brendon Kiernan after Warren Burnell's 81st-minute penalty came back off the post, before Kevin Lokko levelled the scores when his close-range effort after a scramble was adjudged to have crossed the line.
Grimsby substitute Evan Khouri was sent off as the visitors pushed for a winner but it was not to come and although both teams took at least a point from the draw, the rules of the competition mean a penalty shootout was required.
And it was the Mariners who took an extra point after scoring all five of their spot-kicks as Lloyd Kerry hit the post with Harrogate's fifth.