Grimsby fought back from two down to rescue a point with a Danny Rose penalty sealing the result.
Ollie Palmer put Swindon ahead after seven minutes with a deft near-post header which crept in after glancing off the far post from James Ball’s flick-on from a long throw.
Six minutes later, Darren Oldaker found Billy Bodin lurking unmarked at the edge of the box and the striker hammered into the top of the net.
Grimsby halved the deficit when Charles Vernam fired a dangerous low cross in the 22nd minute for Kieran Green to jab in as the Robins backline became muddled.
The Mariners drew level four minutes before the break from Rose’s spot-kick after George McEachran was felled as the visitors piled on the pressure.
Palmer had a golden chance to restore the lead when Princewill Ehibhatiomhan slipped him through, but the veteran striker thumped over.
Ehibhatiomhan then fired at Richard O’Donnell’s legs before the keeper somehow kept out a Will Wright piledriver from 25 yards.
In the dying seconds, Grimsby substitute Jamie Walker volleyed over from 12 yards.
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