Tudgay latched on to a hopeful long ball, wrong-footed the defence and although his shot was saved he followed up with a looping header into the net.
Deon Burton hit the post twice and was also denied by Norwich keeper Paul Gallacher while Ian Henderson twice went close to replying for Norwich.
With three games left the Owls need one more win to be certain of staying up.
Norwich boss Nigel Worthington:
"We created chances, we made tackles and there's a lot of plusses.
"We've had two centre-halves getting tangled up for the goal and we've paid
the price for it. We were good value and a draw would have been fair."
Sheff Wed: Carson, Bullen, Simek, Coughlan, Hills, Brunt (McGovern 68), Adams (Whelan 58), Folly, O'Brien, Tudgay, Burton.
Subs Not Used: MacLean, Best, Spurr.
Booked: Brunt, Whelan.
Goals: Tudgay 45.
Norwich: Gallacher, Spillane (McVeigh 83), Doherty, Shackell, Drury, Hughes (Ryan Jarvis 69), Etuhu, Robinson, Huckerby, Earnshaw, Henderson.
Subs Not Used: Johansson, Colin, Charlton.
Booked: Henderson.
Att: 30,755
Ref: K Stroud (Hampshire).