The Hillsborough side have opened up a seven-point gap from the drop zone and a win at Brighton on Easter Monday will guarantee safety.
Wednesday almost snatched the lead in the ninth minute after full-back Frank Simek found space on the right flank to pick out Chris Brunt.
The Northern Ireland international pumped a deep cross to the waiting Deon Burton whose close-range header was pushed round the post by Paul Gallacher.
The lively Brunt caught out Norwich again as his looping centre reached midfielder Steve Adams who headed back into the goalmouth but centre-back Gary Doherty cleared with Burton waiting to pounce.
Gallacher foiled Wednesday again by tipping Brunt's curling shot over the top before the woodwork saved Norwich. Tudgay flicked the ball to Burton whose shot on the turn beat the Norwich keeper but hit the outside of the near post.
Wednesday broke the deadlock in first-half injury-time when Tudgay kept his cool after his half-volley was pushed out by Gallacher to head the rebound over the stranded keeper.
On-loan keeper Scott Carson kept out Norwich on 55 minutes by blocking Ian Henderson's close-range strike after Michael Spillane's low cross had evaded the Wednesday back four.
Burton had a shot blocked after connecting with a Simek long throw and Brunt pulled a long-range strike over the target as Wednesday tried to kill off Norwich whose play-off hopes ended at Preston last week.
Burton hit the woodwork again with a long-range strike in the dying minutes as the Sheffield side finished the stronger.
The start was delayed as fans and players stood in silence to mark the 17th anniversary of he Hillsborough disaster which claimed the lives of 96 Liverpool fans in the FA Cup semi-final clash with Nottingham Forest.