Danny Ward saved two penalties to send Huddersfield to Wembley on a Premier League dream.
Ward was the Terriers hero as he stopped Sam Hutchinson’s effort on the Owls first spot-kick.
And he came up trumps again to deny Fernando Forestieri as Town won the shootout 4-3.
Wednesday had pushed for a winner in extra-time and on-loan Liverpool goalkeeper Ward made an excellent save from Jordan Rhodes to frustrate the Owls.
Fernando Forestieri then had a great chance, but the Italian’s effort was blocked by the legs of Michael Hefele as both sides edged nervously toward the dreaded penalty shootout.
Wednesday were hit in the opening minutes when Ross Wallace was forced off injured to be replaced by Adam Reach.
Reach had the first decent chance, juggling the ball with his back to goal before turning and hitting a shot, which was saved by Danny Ward, who was back from his one-match ban.
Huddersfield enjoyed more possession and wasted a glorious chance to end their goal drought against Wednesday when Izzy Brown fluffed his lines.
Town swept upfield and Nahki Wells crossed from the left to the near post where the lurking Brown steered his shot wide.
That was a real let-off for Wednesday, who were roared on by the passionate Hillsborough crowd in a raucous atmosphere reminiscent of their play-off thumping of Brighton a year ago.
Mind, the Owls were sloppy too and Steven Fletcher carelessly gave the ball away when Barry Bannan found him in the box with a great through pass.
David Wagner was as animated as ever and moaned to fourth official Stuart Attwell when referee Andre Marriner ignored Huddersfield’s penalty appeal for Keiren Westwood’s challenge on Wells.
The tie badly needed a goal – and it got one from Fletcher on 51 minutes. Bannan sent over a deep cross from the left and the Scot beat Ward to the ball to head home his 11th goal of the season.
Cue pandemonium inside Hillsborough, although the coolest man inside the stadium was Carlos Carvalhal who showed barely any reaction.
Fletcher was feted like a hero when he went off 10 minutes later and the Owls fans must have thought it was time to plan their Bank Holiday trip to Wembley.
But Wagner has shown what a fine coach he is this season and the German made a substitution to change the game.
On went Colin Quaner and 60 seconds later he found Wells, whose scuffed effort went in off Tom Lees on 73 minutes to make it 1-1.
Town pushed for a winner and Westwood made a smart save to keep out Wells’ shot at his near post.