Adam Reach netted twice, while Steven Fletcher also got on the scoresheet, before Wayne Routledge grabbed a consolation strike for the Swans.
There were four changes for the Owls. In came captain Tom Lees, Steven Fletcher, Achraf Lazaar and Rolando Aarons. Out went Jordan Thorniley, Morgan Fox, George Boyd and Fernando Forestieri.
Adam Reach - without a goal in his last 11 games - netted in the 11th minute, drilling the ball home after good work from Lucas Joao.
It was a dominant start for the Owls, who came close to a second when Barry Bannan's cross picked out Fletcher, but the Scotland striker's header was well saved by Kristoffer Nordfeldt.
Wednesday's new left flank of Achraf Lazaar and Rolando Aarons - both on loan from Newcastle United - were having impressive starts.
And the latter was involved for Wednesday's second goal after half an hour, the ball eventually arriving at the feet of Reach, at the far post, and he fired low beyond Nordfeldt.
The Owls had only scored five goals in their previous nine games, but added a third before half-time.
A corner was not cleared, and when Tom Lees hooked the ball back in, Fletcher produced a spectacular overhead kick to make it 3-0. It was the first time this season the Owls had netted three goals in one game.
Nordfeldt denied the pacy Aarons, diving low at his near post, as Wednesday continued their attacking play in the second half.
But Swansea pulled a goal back on 68 minutes, substitute Wayne Routledge seeing his deflected effort beat goalkeeper Keiren Westwood.
The Bannan-Fletcher combination came close to creating another Owls goal, but this time the latter's header failed to test Nordfeldt.
Westwood produced a fantastic save to thwart Courtney Baker-Richardson - after some slack Owls defending invited the visitors back into the game.
And the Owls goalkeeper had to be at full stretch in stoppage time, clawing away a long-range effort from Daniel James.
Owls boss Steve Bruce said: “We made hard work of the second half but considering two or three are sick and not great, two or three are carrying knocks, two in particular haven’t played for months, I knew the second half was going to be a struggle for us.
“I didn’t know it was going to be that big a struggle after the first half but I was delighted with the way we went about it in the first half.
“We were clinical, we went after them with a game plan and certainly worked against them to get after the ball. When you’re getting after the ball and press high like we did, it takes some doing to sustain it.
“Second half left a little bit to be desired, but we were worthy, worthy winners.
“I’m delighted with the way we went about it in the first half. Boyd was off yesterday, Barry Bannan’s not well, Fox hurt his back yesterday so I think we had ten injuries.
“I’m delighted for the two loanees (Achraf Lazaar and Rolando Aarons) and Dominic Iorfa coming on. I think they did terrific on their debuts, which is good to see.”