Mackenzie Maltby nodded home a late winner at Middlewood Road as the Owls Under-21s edged out Burnley by the odd goal in three.
Adam Adetoro gave the Owls a deserved half time lead in the Professional Development League contest but the Clarets drew level 15 minutes from time from the penalty spot.
It was harsh on Wednesday, who had controlled the majority of the contest but just moments after Johnny McDermott fired Burnley level from 12 yards, the Owls were back ahead.
Favour Onukwuli delivered superbly from the right and Maltby climbed high to send a powerful headed effort past Charlie Casper.
Burnley began the game brightly and caused two early scares, with McDermott’s corner clipping the crossbar and Joe Bevan’s rasping drive arrowing just wide of Jack Hall’s near post.
Wednesday composed themselves and worked a way into the game, as Luke Cook’s goal-bound drive was turned away by defender Alex Healey-Byrne in the hosts’ first meaningful attack 25 minutes in.
The Owls continued to press and probe and Adetoro seized his opportunity on 32 minutes from an Owls corner.
Through a crowded box, Casper could only partially punch clear and Adetoro was first to react as the ball landed to stab home from close range.
Wednesday continued on the front foot after the interval as Cook fired wide from Joey Phuthi’s low feed and the lively Onukwuli was stopped just before pulling the trigger in the box after foxing three Burnley defenders in the build-up.
Then 15 minutes from time, Burnley restored parity.
Battling hard to clear the ball, Sam Reed – on the deck – flung out a leg and brought down Joe Ashton in the box. McDermott sent Hall the wrong way to level at 1-1.
It then took just four minutes for Wednesday to regain the one-goal advantage.
Ciaran Brennan threaded the ball down the right for Onukwuli and as the cross came in, Maltby got his timings just right to send a powerful header back across goal and into the net.
It was almost three in the final minute when substitute Bruno Fernandes fed the ball inside for Cook but the striker’s first time shot flew past the near post in the final action of note.
SWFC: Hall, Flannery, Adetoro, Trialist, Brennan, Maltby, Shipston, Reed, Onukwuli, Phuthi (Fernandes 76), Cook (Thornton 89)