The Owls suffered play-off heartbreak as a last-gasp leveller from Patrick Roberts sent Sunderland to Wembley.
Lee Gregory had earlier given Wednesday the lead to wipe out the visitors’ first leg advantage but the Black Cats had the final say.
Wednesday had the better of the game on the balance of play but couldn’t find enough to put the game to bed.
Half chances, mainly for the Owls, punctuated the first passages of play in a rousing Hillsborough atmosphere.
Sunderland, protecting their slender lead from the Stadium of Light, seemed content with attacking on the counter as Wednesday certainly saw more of the ball.
George Byers threatened with an acrobatic bicycle kick that Anthony Patterson gratefully smothered before Gregory headed over the top moments later.
The only other real talking point of a relatively incident-free opening period was on 40 minutes when Josh Windass went down beneath a Dennis Cirkin challenge in the box but the penalty claims were waved away.
The interval provided Wednesday the opportunity to regroup and try to fashion a way through in the second half, the pattern of which matched the first.
Lynden Gooch was mightily relieved to see his interception whistle inches wide of the far post after good work down the right from Windass, with Gregory waiting to pounce.
Patterson dived to gather a Sam Hutchinson effort – before the roof came off Hillsborough as the Owls took the lead on 74 minutes.
Barry Bannan unpicked the lock with a threaded ball to Marvin Johnson, whose slide-rule feed was converted by Gregory.
Wednesday were in the ascendancy and poured forward in pursuit of a second, with Sunderland camped in their own half.
But the one time they did break ranks, the Black Cats scored in stoppage time to seal their route to Wembley.
Jack Clarke motored down the left flank in space and looked up to pick out the onrushing Roberts, who stabbed home from close range.
Wednesday: Peacock-Farrell, Hunt (Mendez-Laing 70), Storey, Dean, Johnson, Hutchinson (Palmer 70), Luongo, Byers (Berahino 90), Windass (Paterson 77), Gregory