After generally playing second fiddle for most of the opening half Grimsby took a firm grip after the break.
They scored twice in five minutes to leave the shot-shy Owls without a hope or prayer.
Former Sheffield United utility man Georges Santos was a colossus in the Grimsby side and his marvellous header in the 48th minute from a corner proved vital.
Santos also headed against the bar by which time the Mariners had the cushion of a two-goal lead.
On-loan midfielder John Oster set up 18-year-old Darren Mansaram and his shot took a deflection before beating keeper Kevin Pressman.
Wednesday boss Chris Turner responded by bringing on two strikers in Adam Proudlock and Leon Knight from the substitutes bench but it made little difference.
Wednesday created and squandered further chances to leave a disappointed Turner asking: "Who is going to score?"We are not good enough in front of goal and we are creating but not finishing the chances.
"The game is all about goals and six minutes of madness cost us this game."For Grimsby, their player-manager Paul Groves limbered up on the touchline but the only change he made was to bring on Jason Gavin in place of right-back Iain Ward near the end.
However, Grimsby held on fairly comfortably for three precious points which keeps them in touch with Bradford City who are just above them in the table.
This victory means that the Mariners have won all the vital matches against their companions in distress following 2-1 wins at Brighton and Stoke.