The Blades took the lead just three minutes into Sunday's game when David Brooks won a free kick 25 yards out, which was ferociously driven home by John Fleck.
United then doubled their advantage on the quarter-hour mark courtesy of Leon Clarke, who got behind the napping Owls defence, latched on to Enda Stevens's long ball, and poked beyond Keiren Westwood.
Wednesday halved the deficit right on the stroke of half time, with the ball played forward into the box for Rod Wallace, who passed into the middle for Gary Hooper to finish at the near post.
The hosts then appeared to have sealed a comeback in the 66th minute when Adam Reach powered forward and crossed low for Lucas Joao to control and then rifle a shot past Jamal Blackman.
Wednesday's resurgence was short-lived, however, because just one minute later, Mark Duffy collected Clarke's clever pass over the top before smashing a shot past Westwood at his near post to put Chris Wilder's side ahead again.
The game was then put beyond the Owls' reach through yet more comical defending as, 77 minutes in, Clarke managed to bundle between two defenders, win a ball over the top, and slot beyond Westwood.
The result sees United edge into the Championship's playoff places, sitting fourth after nine games, while Wednesday remain 10th.