The teams kicked off backed by an electric atmosphere unlike anything previously heard this season.
The Owls were arguably the most fired-up and James Quinn came close in the sixth minute with a looping header that keeper Boaz Myhill had to tip over.
The opening 15 minutes were fast-paced but neither side could carve out any clear-cut chances until the 19th minute when Andrew Talbot broke the deadlock for the Owls.
Jon-Paul McGovern had put an excellent ball in and despite City's attempts to clear off the line Talbot bundled the ball in from just yards out.
City went on the attack straight away with Owls keeper Paul Gallacher saving Kevin Ellison's header following a great cross from Stuart Elliott on the left wing.
Craig Fagan had a shot from the byline saved in the 36th minute but the Tigers were losing their bite and the massive home crowd were beginning to get audibly frustrated.
Just two minutes before half-time Gallacher fumbled in the box and there were a few moments of chaos before the Owls defenders could clear the danger.
Minutes after the restart McGovern was inexplicably left in acres of space and allowed to shoot at goal from 20 yards but Myhill did well to beat the shot away.
The second half then exploded into life when Gallacher produced a superb save to deny substitute Andy Hessenthaler, before Lee Bullen was ruled to have handballed in the area.
Despite protests from the Owls defenders, the league's top scorer Elliott stepped up and placed a low shot past Gallacher into the bottom right-hand corner to notch up his 29th of the season in all competitions.
There was a fantastic reception for Tigers substitute Ben Burgess, who was introduced in the 73rd minute to a standing ovation.
Burgess had been cruelly injured at the climax of City's last promotion and has been sidelined for the best part of a year.
But Wednesday continued to put the pressure on and City had a few dangerous moments near their goalmouth before Steve Adams saw his effort well saved in the 75th minute.
And the Owls dramatically grabbed an injury-time winner when Quinn bundled in from one yard out.