Grimsby had the first chance after three minutes. Paul Bolland broke through the middle, but his disappointing shot was saved easily by Conrad Logan.
Michael Raynes replied for Stockport with a near-post effort which was blocked, then on 17 minutes Paul Turnbull's 25-yard shot was similarly beaten away.
The match finally came to life in the 25th minute when the visitors took the lead against the run of play.
Nick Hegarty swung in a great cross from the left and Martin Butler tucked home his first Mariners goal from eight yards.
Stockport responded with Adam Nowland squeezing a near-post shot just wide, but at the other end Butler almost doubled his tally on 37 minutes when his diving header was saved by Logan.
Grimsby came mighty close to a second goal five minutes after the restart. Nick Fenton's header appeared destined for the bottom corner, but Stockport skipper Gareth Owen cleared the ball off the line.
Centre-back Ryan Bennett then fired over for the visitors, but Stockport were still probing and on 61 minutes Raynes headed inches wide from Rose's out-swinging cross.
Stockport frontman Liam Dickinson saw a shot deflected over the crossbar, before the hosts did level with just eight minutes left.
Sub Anthony Pilkington was chopped down just outside the box and up stepped Ros,e who curled in a great free-kick past the despairing dive of Phil Barnes.