Things went from bad to worse 11 minutes from the end when Andy Hughes was dispossessed when juggling the ball. It was played forward by Wednesday and Chris Brunt sent the Owls fans wild with delight as he poached the winner from close range, after Robert Green could only block a shot by substitute Ritchie Partridge.
The Owls started well and Brunt's low left wing cross was turned onto the near post by Gabriel Agbonlahor before Green managed to grab it.
However, the hosts soon hit back and Wednesday twice cleared off the line to deny Dean Marney and Youssef Safri.
Marney then released his skipper, Hughes, but he blasted over with only the goalkeeper to beat.
The wasted chances continued to come and, after 33 minutes, Dean Ashton beat the offside trap and David Lucas, but saw his shot drift wide.
Five minutes later Ashton did get a shot on target but Lucas made a comfortable save.
The Canaries continued to press after the break and shortly after the hour mark Darren Huckerby's low shot was saved by Lucas. And five minutes later Ashton was inches away with a rasping drive
The hosts thought they would get a great chance to break the deadlock when Craig Fleming was bundled over in the Owls penalty area, but the referee gave a goal kick.
Lucas did well to keep out another fierce shot from Ashton.
Then, in the 79th minute Hughes was guilty of giving the ball away and cost Norwich the match as Brunt fired in the rebound after Green parried Partridge's shot.