It was Grimsby who took the lead in the 14th minute when a sweeping ball from the right by John Oster allowed Steve Livingston to beat the offside trap and chip the ball over Neil Cutler from just inside the box.
The Potters could have easily pulled it level in the 21st minute when a cross whipped in from Clive Clark fell on to the head of Greenacre but he got little direction and the ball wastefully went wide of Danny Coyne's goal.
Stoke were caught on the back foot in the 22nd minute when a foul by Sergei Shtaniuk gave Stuart Campbell a free-kick who quickly took it and took it into Cutler's bottom left corner.
A brilliant save by Coyne denied Stoke again when a Peter Hoekstra corner floated to the back post where Tommy Mooney powered the ball at the keeper's top right corner but he proved every bit equal.
The scoreline didn't particularly reflect the play on the pitch going into half time but to even grab a point Stoke needed to capitalise on their chances.
Stoke responded well four minutes into the second half when a handball in the box by Jason Gavin gave them a penalty which Mooney subsequently slotted into Coyne's bottom right corner.
Grimsby managed to quickly break in the 64th minute when Paul Groves spotted Oster on the right and supplied him with a ball that allowed him to get into the box with his low drive which was saved by Cutler and cleared to safety.
The Mariners rarely managed to push into Stoke's half during the last stages of the game with the Potters applying crushing pressure but reaping no results.
Grimsby boss Groves said: "It's nice to have come away from a game and not have played particularly well but get all three points.
"I wanted more from the team but we made it difficult for them to break us down, it's a good boost and a nice little run we're on now.".