The Welsh side fielded four new signings while Grimsby recorded their fifth win on the trot without ever approaching top form.
The only goal in a scrappy match came on 68 minutes.
A probing ball down the Grimsby left from skipper Tom Newey was latched on to by leading scorer Danny North.
He flicked it over keeper Gavin Ward and, as it ran towards goal, a Wrexham defender cleared but one of the referee's assistants clearly flagged to show it had gone over the line.
It was no more than Grimsby deserved for they had carved out most of the chances, hitting the woodwork through midfielder Paul Bolland in the first half and then again from sub Justin Whittle near the end.
Wrexham's one defining moment was a shot from lively striker Paul Hall, one of the newcomers, which was brilliantly saved by Grimsby keeper Phil Barnes.
Wrexham did ride their luck at times but this was an encouraging show for manager Brian Little and the 172 travelling fans.
But Grimsby had the chances to have won more comfortably with striker Gary Jones and wing-back Nick Hegarty going close in the first half.
And in a dramatic second half both teams had a player stretchered off and referee Graham Salisbury had to be replaced by assistant M Brown as he was injured.
Because of this there were six minutes of stoppage time which meant an anxious last few minutes for the home fans who were determined to cheer their favourites over the line.
Grimsby are now moving up into mid-table and, with an away game to come at Dagenham and Redbridge next week, they will be hopeful of adding to their points tally.