The Hillsborough side, whose last victory against United was in September 2002, struck in each half with both sides missing chances after the interval as the game opened up.
Wednesday, with just one win from seven games, found a gap in the United rearguard early on as Tudgay ran onto a Jermaine Johnson pass with the Wednesday striker over-hitting his shot as the ball flashed over the target from the edge of the area.
United were on the back foot as dangerman Johnson almost grabbed the opener in the 17th minute when he drilled a right-foot shot which Paddy Kenny finger-tipped over the bar.
The breakthrough came in the 25th minute when Johnson was given too much space by Chris Armstrong as he swept over a low cross.
The ball curled away from the diving Kenny as Akpo Sodge side-footed it over the line for his seventh of the season.
United were struggling to get into the game but had a half chance when Manchester United loanee Lee Martin picked out Michael Tonge 12 yards from goal, but the midfielder mis-kicked and the ball was hacked clear.
The game began to open up as a mistake by Mark Beevers handed United a chance to draw level.
The centre-back was dispossessed by David Carney who picked out Tonge, but he pulled his low strike wide of the far post.
United survived as Burton O'Brien lashed a spectacular shot over the target after a neat back-heel from loan signing Ronnie Wallwork, before Sodje nipped in as Leigh Bromby failed to head clear his low effort.
Tudgay almost snatched a second deep in first-half injury time. Sodje connected with Glenn Whelan's lofted ball into the goalmouth and the little striker produced a spectacular overhead kick which was pushed around the foot of the post by Kenny.
Keith Gillespie and Billy Sharp replaced Carney and Nick Montgomery as Bryan Robson injected more firepower into his mis-firing side.
United took the initiative and were foiled by Whelan clearing Rob Hulse's close-range header off the line after Wednesday had failed to clear Martin's cross.
The Hillsborough side squandered a chance in the 69th minute as Johnson and Tudgay exchanged passes to let in substitute Leon Clarke, but the big striker hit a lazy shot straight at the United keeper.
Wednesday added the second 14 minutes from time as Johnson's long throw reached Clarke and he picked out Tudgay, who volleyed into the roof of the net from six yards.