With the tie evenly poised at 0-0 from the first leg, there was concern early on for the home fans when Ross Wallace went down clutching his hamstring and was substituted just five minutes into the game, Adam Reach taking his place.
However, the Owls were the brighter side in the early exchanges as Terriers goalkeeper Danny Ward was tested twice in the 20th minute, keeping out Reach's close-range shot and then the follow-up from Steven Fletcher, who was offside.
The Terriers also had a legitimate shout for a penalty turned down seven minutes before the break as Wells appeared to have been brought down by Owls stopper Keiren Westwood, but referee Andre Marriner waved away the appeals.
Wednesday went on to take the lead five minutes after the restart when Barry Bannan's pinpoint cross found the head of Fletcher, who made no mistake nodding home to send the home crowd into a frenzy.
However, David Wagner's charges drew level 73 minutes in when substitute Collin Quaner whipped a cross in from the right and Wells turned the ball past Westwood from close range, with the score level upon the full-time whistle.
Neither side could be separated in extra time, forcing the game to be decided by spot kicks and, although Jack Payne missed for the Terriers, Sam Hutchinson and Fernando Forestieri fluffed their penalties to put the underdogs just one game away from the Premier League.