With new manager Chris Powell watching from the stands, the Addicks went ahead in the fourth minute when Therry Racon's cross from the left was headed down by Christian Dailly for Scott Wagstaff to fire home from eight yards.
Two minutes later, Owls skipper Darren Purse hauled down Joe Anyinsah inside the box and top scorer Johnnie Jackson sent Nicky Weaver the wrong way with a spot-kick for his 14th goal of the season.
Wednesday blew a chance to hit back when Reda Johnson was pushed by Gary Doherty in the area in the 38th minute but keeper Rob Elliot dived full length to keep out Tommy Miller's penalty.
The double substitution of Paul Heffernan and Jermaine Johnson paid off almost immediately when a free-kick was not cleared properly and Heffernan lashed home the loose ball from eight yards in the 47th minute. And after Anyinsah had missed a sitter, Chris Sedgwick's left-wing cross was headed in by Johnson for his first Owls goal on his home debut.
As Wednesday pushed for the winner, Heffernan's header hit the post in the fourth and final minute of stoppage time.