The sides started the game in fourth and fifth in League Two, with the hosts knowing that only a victory would be good enough to keep the pressure on Northampton and Leyton Orient.
After a slow start Grimsby took a grip on proceedings and went close through Curtis Woodhouse.
A tremendous run from Michael Reddy set the midfielder up, but he got under his shot and blazed over the top.
Woodhouse then forced a good save from Robins keeper Shane Higgs, with Reddy putting the rebound into the side netting.
Grimsby knew the chances would come after the break if they were patient, but also that they had to avoid any lapses at the back.
Two minutes into the second half, however, the Mariners almost conceded when goalkeeper Steve Mildenhall spread himself to deny Steve Guinan from no more than two yards.
After that Grimsby took control again and the vital goal came after 69 minutes.
From a free-kick midway inside the Cheltenham half Woodhouse floated the ball over and Gary Jones stole in to head home is 14th goal of the season.
After that it was Cheltenham who had to come forward and skipper John Finnigan saw a shot beaten away by Mildenhall.
Visiting boss John Ward threw on sub John Spencer alongside Kayode Odejayi to give the Robins a more potent strike-force.
However, the crowd played their part by roaring Grimsby on and they negotiated the anxious last few minutes to secure a crucial victory.