All three goals came in an electrifying first half with Italian striker Arturo Lupoli grabbing United's goal after the Hillsborough side had swept into the lead
Wednesday got off to a shock lead after just 49 seconds as midfielder Michael Gray pumped in a low cross and Spurr charged through unmarked to beat Paddy Kenny with a low strike from 14 yards.
The lead lasted just three minutes after United won a throw in deep in Wednesday territory.
Greg Halford looped the ball high into the goalmouth as Lupoli stretched to connect with his header going in off the underside of the bar.
Another Halford throw caused more problems but Lupoli's overhead kick was blocked. The ball fell to ex-Wednesday defender Leigh Bromby, who finished with a right-foot shot which took a deflection and flew inches wide of the target.
The Hillsborough side regained the lead on 29 minutes with Matt Kilgallon clearing straight to Tudgay, who raced through unchallenged to lash in an unstoppable shot from 25 yards into the top left hand corner.
Wednesday stepped up the pressure with Tudgay heading on a long throw to Leon Clarke whose right-foot volley from 10 yards cannoned off the post with Kenny stranded.
Jamie Ward, on for Lupoli, was inches away from grabbing an equaliser in the 68th minute as the little striker connected with Halford's cross but his close-range heading looped the wrong side of the bar.
Gray foiled Ward with a last-ditch tackle six yards out as the former Chesterfield striker was about to shoot and substitute Billy Sharp's low cross was steered clear by Lewis Buxton.
Sharp's shot on the turn was tipped over late on by Lee Grant as Wednesday held on.