Jermaine Johnson tormented the Blades in the first half and Akpo Sodje scored the Owls' opener from his low cross.
Blades keeper Paddy Kenny also made two superb stops to keep out another Sodje shot and a Tudgay overhead-kick.
United dominated after the break but, seconds after they had Billy Sharp's goal ruled out for offside, Tudgay met Leon Clarke's cross to make it 2-0.
Sheffield Wednesday boss Brian Laws:
"This win is a really special win for me and even more special for the fans who have been very long suffering.
"The victory has given back some pride. It was a tremendous win and we will take a lot out of it.
"The second goal gave us a little bit of breathing space and they looked as if they gave up."
Sheffield United boss Bryan Robson:
"We've still got a chance of getting into that top six, but we're not kidding ourselves.
"We're over halfway in the season now and we felt things were coming together nicely, but maybe we're not good enough to get up there.
"Maybe the expectations at the start of the season were too much for the players to take on board because the league table doesn't lie."
Sheff Wed: Grant, Bullen, Beevers, Hinds, Spurr, Whelan, O'Brien, Johnson (Small 79), Wallwork, Tudgay, Sodje (Clarke 62).
Subs Not Used: Burton, Wood, Watson.
Goals: Sodje 25, Tudgay 76.
Sheff Utd: Kenny, Bromby, Kilgallon, Armstrong, Geary, Carney (Sharp 46), Speed, Montgomery (Gillespie 46), Tonge, Martin, Hulse.
Subs Not Used: Naysmith, Stead, Ehiogu.
Att: 30,486
Ref: Mark Halsey (Lancashire).
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