The striking duo, who were team-mates at Derby County, linked up superbly as Tudgay grabbed a brace of goals to maintain Wednesday's amazing revival under caretaker boss Sean McAuley.
But McAuley, the club's Academy director, is set to be dropped on Monday with Scunthorpe boss Brian Laws waiting in the wings to take over.
McAuley has steered Wednesday to three wins and a draw since Paul Sturrock's shock sacking by multi-millionaire casino boss Allen.
Wednesday forced their way in front with a piece of sparkling football that gave Tudgay his fourth goal in as many matches.
Midfielder Kenny Lunt swung in a right-wing cross that Tudgay headed onto strike partner Burton who cushioned his header into Tudgay's path and he smashed in a first-time shot from ten yards.
The advantage boosted the Owls' confidence and they could have gone further in front before half-time but Yoann Folly cut in from the left and drove a shot straight at the keeper.
But five minutes before the break Wade Small pounced to rob Darren Kenton and went darting into the area before unleashing a 15-yard shot that brought an impressive save from Conrad Logan.
And Wednesday put the result beyond City early in the second half when they added a second goal to leave the visitors floundering.
Ten minutes after the restart it was Tudgay and Burton who combined again to stretch the Wednesday lead with another well-worked move.
Burton raced to the byline on the left, spun cleverly and drilled in a low cross as Tudgay shook off his marker and rammed home a first-time left-foot shot from 12 yards.
Three minutes later Tudgay was unfortunate not to claim a hat-trick when his 25-yard effort through a crowd of players caught out Logan but his shot drifted just wide.
Tudgay's flick then sent Burton tearing goalwards and the striker smashed in a brave first-time effort that was just over.
But after 72 minutes a blunder by the otherwise impressive Madjid Bougherra let in City. He headed a clearance straight up in the air and Matty Fryatt cashed in, smashing a ten-yard shot past Bradley Jones.
However, the Owls hung on to claim the three points as they moved further away from the bottom three.