Jordan Cook was shown a second yellow card after 33 minutes, but the Hatters weathered a decent spell from Grimsby towards the end of the first half, before James Collins headed in after Luke Berry's free-kick had been tipped onto the bar in the 49th minute.
The Hatters saw out the result well as Grimsby pushed for an equaliser, meaning Jones' side are now nine points clear at the top of Sky Bet League Two.
Jones said: "It's a massive win, a huge win in the context of everything. It's an absolutely massive win. The way we have dug in and grafted after going down to ten men.
"We said we were going to get one opportunity and we're going to have to take that opportunity and we did. Then we showed a real grit, a desire to defend our box. It's all about results these days, it wasn't pretty in anyway from us, but I would have taken that all day long.
"Today was a great away performance, I probably wouldn't have it any other way. They don't half galvanise you these sorts of results.
"Let's talk about the ten that really dug in. Lawson D'Ath came on and really impacted in everything he did and the team did do that. Let's talk about that because it's a really proud day for me. To dig in, the fans came up and got behind us. We dug in and got a goal and then just defended."