On-loan midfielder Michail Antonio fired home at the death after James Berrett had cancelled out Chris Lines' opener.
In an open first half, Francois Zoko tested Owls goalkeeper Stephen Bywater at the midway point before Adam Collin was forced to tip Danny Batth's header over at the other end.
Lines put the home side ahead with a scuffed finish into the bottom left-hand corner in the 26th minute after Collin could only parry Jermaine Johnson's effort.
Wednesday should have extended their advantage seven minutes before the break when Johnson broke clear down the right but the Jamaican blasted straight at Collin instead of setting up Gary Madine for a simple tap in.
Berrett levelled the contest nine minutes from time with a right-foot shot into the bottom right-hand corner from the edge of the area before Antonio cut in from the left to lash the ball past Collin.