Adams' men moved up to 10th place in the table after strikes from Louis Dodds, Tom Fraser, Gareth Owen and Lee Collins, while Marc Richards had his penalty saved by Grimsby debutant Josh Lillis after Danny Boshell was sent off for a professional foul on Lewis Haldane.
Adams said: "It's a long time since Port Vale fans have had so many smiles on their faces and we certainly don't want to take anything away from that.
"But we do have to keep taking little strides in the right direction and make sure we don't slip backwards.
"I'm delighted with the application of the players and we must be doing something right because we are winning games of football. They are working very hard and taking on board what we want them to do.
"Grimsby came here to take their time with everything and slow the tempo of the game down, but the first goal settled us and after that it was another good day at the office."