But once midfielder Paul Bolland had scored with a picture-book header from a Tom Newey cross on 51 minutes it was Grimsby who bossed the game and looked the likeliest winners.
Just after the equaliser Grimsby boss Alan Buckley took off striker Andy Taylor replacing him with midfielder James Hunt, to go 4-5-1 which was greeted with boos by many of the crowd. Soon after another substitute, striker Peter Bore joined the fray as Grimsby switched back to 4-4-2.
In the first half, Matty McNeil had a header ruled out for off-side for the visitors. He then grabbed the opener just before the break when he was played in by substitute Anthony Elding and burst past two defenders to poke the ball past keeper Phil Barnes.
This was probably rough justice on the hosts, but they reacted well after the break and in the end were at least full value for their fourth one-one draw in six matches. In fact this was only Grimsby's second point in their last six meetings with the Cheshire side.
It was mostly Grimsby on top in the closing stages and with almost the last kick of the game, striker Isaiah Rankin had a shot kicked off the line by Stockport skipper Ashley Williams.
It was terrific defending and his team-mates could be thankful to their skipper for saving the away point.
It was a good away point for County who have yet to win on their travels this season and this is a lucky ground for them as they had won on their previous two visits.