Having led just after the break through Marco Matias' opportunistic effort, Wednesday then collapsed and conceded twice inside a minute.
The defeat means the Owls have just one win in their past ten games.
Wednesday started well and Lucas Joao was put through inside the opening ten minutes but took too long to pull the trigger when well-placed.
Swansea enjoyed plenty of the ball throughout the first half but were limited to chances.
The best saw Wayne Routledge slipped in but the winger placed a shot just wide of Cameron Dawson's post.
Joao spurned a similar chance for the Owls when well-placed but his shot slid wide of goal and did the same just after the restart.
Just after the hour though, the Owls took their chance and made it 1-0.
Sub Matias, a half-time replacement for Josh Onomah, picked the pocket of a Swansea defender and coolly slotted home.
Just when the Owls looked in a strong position they suddenly fell apart and conceded twice inside two minutes.
First, sub Bersant Celina fired home after good work down the right from the hosts.
Then barely 60 seconds later Wayne Routledge outpaced Morgan Fox and his shot went through the legs of Dawson.
Swansea looked dangerous from the counter and Oli McBurnie's header came back off the woodwork as the hosts pushed for a third.
Wednesday couldn't muster something in the final stages with the hosts clearing their lines after a succession of corners to consign the visitors to yet another defeat.
Swansea: Mulder, van der Hoorn (Carter-Vickers 46), Naughton (Celina 66), Rodon, Roberts, Grimes, Fer, Dyer (Montero 63), Routledge, McKay, McBurnie
Subs not used: Nortfeldt, Asoro, John, Fulton
Owls: Dawson, Lees, Pudil, Thorniley, Baker, Pelupessy, Onomah (Matias 46), Reach, Fox (Fletcher 79), Joao, NuhiuÂ Â
Subs not used: Wildsmith, Palmer, van Aken, Jones, Hector