In a game of few chances, Kabongo Tshimanga struck the winner with six minutes remaining after Fraser Kerr headed down Jim Kellermann’s free-kick.
The win means Town stay top, are 12 unbeaten and have lost one in 21 in all competitions despite an ongoing injury crisis.
"We make no excuses for the scrappiness of the game,” manager James Rowe said.
"It was the same at Salford, we are finding a way to win. The team values are at the forefront of that.
"It is remarkable. We have got 11 out and we are still picking up results and getting clean sheets.
"I don't think Scott Loach had a save to make so we were well worthy of the victory.”
In the first-half both teams had one big chance each. Grimsby’s Harry Clifton headed wide when unmarked in the box and up the other end Kellermann was denied by a brilliant save from Max Crocombe.
"I think in both halves we were patient and as the halves grew on we became stronger and looked the most likely,” Rowe said.
"I think their goalie has pulled off a really good save in the first-half from Jim when we were in the ascendency. It was a good move and a wonderful save.”
With the clock ticking down, Tshimanga fired in the winner for his 19th goal of the season.
"It is a good finish from Kabby and that is all he needs,” Rowe explained.
"I was telling him to keep going with 20 minutes to go.
"Yes, it had not been his day before that perhaps. He was getting a lot of physicality from their back three but he never stopped running and he deserved that chance and that is all he needs. It was a fantastic finish on the drop down.”
On the overall unbeaten run, Rowe added: "They set the values and the standards every day.
"It has been key how we have bounced back from Woking.
"We want to make sure if we have a bit of adversity - we are going to lose some stage - that we bounce back quickly."