The Owls beat Burton on penalties in the first round of the FA Youth Cup following a 1-1 stalemate at Hillsborough and repeated the feat against Barnsley to advance to round four.
Pierce Charles was the hero, saving two spot-kicks, as the hosts emerged triumphant.
Reds goalkeeper Rogan Ravenhill saved from Rio Shipston, who had given Wednesday an early lead in normal time but then gave the penalty away from which Fabio Jalo equalised at the start of the second half.
Charles then kept out Rafael Montiero in the shootout before Jonathan Bland hit the bar. Charles saved next from Charlie Hickingbottom, leaving Sean Fusire to slam home the winner from 12 yards.
The Owls were almost off to the perfect start in the regulation 90 but for a stunning save from Ravenhill.
Jake Bradford delivered a low feed from the right flank, Jarvis Thornton got a sweet connection 12 yards out and Ravenhill flew through the air to tip over the top.
Wednesday went on to dominate and after a period of pressure made the breakthrough 20 minutes in.
Shipston curled an inviting set piece right into the danger zone and the ball evaded a mass of bodies in the box to go straight in at the far post.
Barnsley did fashion a response without troubling a well organised rearguard and half time came with the Owls in charge.
But that changed just a minute into the second period as the Reds levelled from the spot.
Shipston was adjudged to have fouled Jalo in the area and the latter picked himself up to send Charles the wrong way.
Jalo went close on the hour to doubling his tally, planting a firm header goalwards only to see Thornton in the right place on the line to clear.
Charles scooped a cross-cum-shot from Mylan Benjamin out of his top corner before the contest developed into a scrappy affair, with much of the play in the engine room.
Neither goalkeeper was troubled in the closing stages, but for Ravenhill making a routine near post save from Bailey Cadamarteri – so to extra time the game went.
With tired legs and plummeting temperatures, there was little action of note, save a Devlin Moses burst that pulled Ravenhill from his goal to deny the pacey forward.
The tie, though, needed a winner on the night and the penalty shootout saw joy for Wednesday and heartbreak for the Reds.
Wednesday: Charles, Bradford (Johnson 100), Flannery, Maltby, Chapman, Tapudzai, (Moses 46), Fusire, Shipston, Cadamarteri, Thornton (Fernandes 106), Phuthi