It was arguably the Mariners' best scalp of the season as the Robins are pushing hard for promotion from League Two.
In a match of remarkably few chances the decisive strike came on 73 minutes. It came about following a splendid advantage decision when veteran full-back John McDermott was fouled as he made progress down the Grimsby right.
Play was waved on and midfielder Ciaran Toner beat Swindon keeper Phil Smith from 15 yards. It was the midfielder's eighth goal of the season which made him Grimsby's joint top scorer.
But this was a real team effort by Grimsby, who defended so soundly that the visitors had only one clear-cut shot on goal and that did not arrive until ten minutes from time.
The dangerous Christian Roberts made space for himself and hit a fierce shot which was smartly turned round the post by keeper Phil Barnes.
The whole Grimsby team defended splendidly with skipper Justin Whittle and fellow central defender Nick Fenton a solid barrier at the back.
The game also hinged on two astute substitutions by Mariners boss Alan Buckley.
Firstly with the 4-5-1 formation not working he took off midfielder Danny Boshell with striker Gary Jones giving Danny North much needed support upfront.
And then for the second half winger Peter Till came on for Peter Bore and he had a good match down the right.
After going behind Swindon tried desperately to pull things round, but their cause was not helped when midfielder Michael Timlin was sent off on 81 minutes.
Booked six minutes earlier for a foul he repeated the action and the referee had no choice but to show him first another yellow and then a red card.
But Grimsby's defence, which has not been the best all season, held out for a third successive clean sheet despite Swindon forcing successive corners as the three minutes of added time drew to a close.