Steve Guinan netted the crucial goal just after the hour mark with a low cross that evaded everyone and crept into the bottom corner of Steve Mildenhall's goal.
However, Cheltenham deserved their luck as they played the better football and created the clearer chances.
One of those chances led to a 71st-minute spot-kick when Curtis Woodhouse felled Grant McCann in the box. McCann got up to take the kick himself but saw it pushed away by Mildenhall.
Grimsby could not make the Robins pay for the squandered opportunity and John Ward's men were able to hold on despite a late spell of pressure.
Cheltenham started the better and Steven Gillespie and Guinan both wasted good chances inside the first 10 minutes, with Mildenhall easily saving the two tame shots.
The Mariners lost their main attacking threat when Michael Reddy was forced off following a clash of heads with Craig Armstrong, who was also forced to leave the action.
They recovered from the setback to enjoy their best spell of the game in the closing stages of the first half, with Robins keeper Shane Higgs called into action to push away a low shot from Gary Cohen, and Andy Parkinson and Paul Bolland both going close with shots from the edge of the box.
Grimsby were unable to keep the momentum going after the break and McCann took control of midfield for Cheltenham.
However, they struggled to create chances and it was in keeping with their play that the goal arrived from a cross that rolled straight in.
McCann should have made certain of victory from the penalty spot eight minutes later, but his miss did little to dent his confidence as seconds later he attempted a wonderful chip from the edge of the area that struck the crossbar.
The rebound fell to Brian Wilson but Mildenhall somehow made up the ground across his goal to keep the midfielder's effort out and the game alive.
The Mariners threw everyone forward and almost got a reward when Junior Mendes' cross was met by Gary Jones, but Higgs reacted sharply to beat the ball away and preserve Cheltenham's advantage.
The Robins should have made it two when Guinan and Kayode Odejayi broke in the closing minutes, but Odejayi was reluctant to shoot and the opportunity came to nothing.
Minutes later it mattered little as the referee's whistle prompted wild celebrations from the Cheltenham camp.