The visitors dominated the match with Aaron Ramsey hitting the post, Gavin Rae heading just over and Jimmy-Floyd Hasselbaink keeping Lee Grant busy.
But Tudgay's eight-yard strike against the run of play four minutes from the break was enough for the three points.
Cardiff have now gone four games without a win in the Championship.
Cardiff boss Dave Jones:
"We should have won the game. They've not caused us any problems today whatsoever. Even the goal we should not have conceded and that's lack of concentration at the back. "We'd made the game easy by working really hard and you've got to keep the backdoor shut, but that's what happens at all levels of football.
"We gave them hope and we gave them something to hang on to, which they did."
Sheff Wed: Grant, Bullen (Hinds 26), Gilbert, Beevers, Wood, Kavanagh, Johnson (Sahar 73), Small, Bolder, Showunmi (Burton 86), Tudgay.
Subs Not Used: O'Brien, Wallwork.
Booked: Johnson.
Goals: Tudgay 41.
Cardiff: Enckelman, McNaughton, Capaldi, Purse, Johnson, Rae, Ramsey, McPhail (Thompson 74), Whittingham, Hasselbaink, Parry.
Subs Not Used: Oakes, Sinclair, Brown.
Booked: Purse.
Att: 18,539
Ref: Darren Deadman (Cambridgeshire).
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