Wednesday surged into a three-goal lead, including a superb 40-yard effort from Chris Brunt, but Leeds almost snatched a point with two goals in the final two minutes.
Both teams were without their captains. Leeds lacked Kevin Nicholls, who unbelievably told manager Dennis Wise that he wants to return to Luton, while Wednesday gave Graham Coughlan compassionate leave after his younger brother Shane was found drowned in a canal back home in Ireland.
Leeds brought in Bulgarian international midfielder Radostin Kishishev, who had joined them on loan from Charlton 24 hours earlier.
Wednesday got off to a perfect start with a goal after only seven minutes. Deon Burton's downward header bounced up against the bar and the ball was cleared out to the left.
Tommy Spurr sent over a perfect cross and Tudgay's header into the far corner gave goalkeeper Graham Stack no hope.
Wednesday keeper Iain Turner, on loan from Everton, made splendid saves from Ian Moore and David Healy before Healy felt he should have had a penalty when he stumbled under Steve Watson's challenge.
Leeds paid the price for failing to take their opportunities and a bad mistake by Frazer Richardson was punished by Brunt for Wednesday's second goal in the 37th minute.
Richardson dallied on the ball 40 yards out near the touchline and Brunt robbed him, spotted Stack off his line, and cheekily lobbed the keeper for a magnificent goal.
Leeds started the second half with Robbie Blake in place of Jonathan Howson as they gambled everything on attack. Their hopes rose when Tudgay was sent off for his second bookable offence - kicking the ball away.
It brought a frown to the face of manager Brian Laws who felt the punishment was too harsh but Laws was smiling again in the 54th minute when Wednesday grabbed their third.
Burton and Glenn Whelan combined on the left and the ball was played inside for former Bradford City winger Jermaine Johnson to shoot powerfully past Stack.
Wednesday have been noted for conceding soft goals this season and when Lee Bullen sliced the ball into his own net two minutes from the end there seemed little for the Owls to fear.
However, in a nerve-jangling finish, Richard Cresswell crashed in an overhead shot as the game entered stoppage time as Leeds threatened to claw their way back for a draw.