James O'Connor scored the decisive penalty for Wednesday after Billy Clarke and Neal Eardley put Blackpool on the back foot by missing their first two kicks.
The winners' reward is a trip to face Premier League side Blackburn Rovers in round two.
Blackpool were made to pay for a string of missed chances, but both sides were guilty of woeful finishing.
The Championship side, who had Ashley Eastham sent off for a horrific tackle on Mark Reynolds, controlled the majority of the game. Clarke and Craig Sutherland in particular missed golden chances in the opening 90 minutes.
However substitute Tom Barkhuizen almost won it for Blackpool with six minutes left, leaving a trio of defenders in his wake before forcing home goalkeeper Richard O'Donnell to palm his hard strike on to the crossbar.
Blackpool made 10 changes from Friday night's opening league win over Hull, but League One club Wednesday also fielded a weak side as they face Bournemouth on the south coast in less than 48 hours.
The Owls' players scored all of their penalties, Liam Palmer, Jon Otsemobor and Clinton Morrison giving them a 3-1 lead with Ludovic Sylvestre replying to keep the Tangerines in it.
Miguel Angel Llera then made it 3-2 - but O'Connor smacked his kick home to see Wednesday through.