Peter Sweeney earned the Mariners a deserved point with a 67th-minute equaliser after Paul Mullin had stolen his side the lead with 60 minutes gone.
Grimsby dominated the first half and went close on three occasions. Sweeney saw a curling free-kick tipped round the post by Barry Roche.
Then Michael Coulson cut in from the left and saw his shot from 20 yards pushed away from the keeper. And from an in-swinging Sweeney corner Rob Atkinson’s near-post header dropped just past the far post.
Morecambe took the lead against the run of play when Mullin fired home with his right foot from 12 yards after a Garry Hunter cross had been half cleared.
But Grimsby equalised when a clearance was picked up by Sweeney who fired a dipping volley into the top corner.
Friday night’s other scheduled League Two match between Hereford and Lincoln at Edgar Street was postponed due to the weather.