The pressure looked like it was going to be lifted on the Owls boss when Marco Matias stole the ball from defender Joe Rodon and found the back of net after 63 minutes.
But the introduction of Bersant Celina on 66 minutes turned the game in Swansea’s favour.
The former Manchester City academy graduate produced a composed finish to level the scores after 71 minutes after Barrie McKay slipped Connor Roberts in behind the defence to make the final pass.
Then less than a minute later he picked up the ball in the middle of the pitch and lifted an inch-perfect pass over the defence for Routledge to chase.
The 33-year-old took the ball in his stride and fired a low shot through the legs of Cameron Dawson and into the back of the net to complete the comeback and heap more pressure on Wednesday manager Luhukay.
“It’s very hard to take, it is totally frustrating to give away a 1-0 lead in a couple of minutes,” said Luhukay.
“Swansea had no chance to score, but we didn’t have any discipline or concentration to hold the lead.
“In a couple of minutes we gave everything away. We didn’t defend down our left side twice and that is not acceptable, you have to run with your opponent.
“I have tried to coach this three or four times, but it is not going in their heads.
“I am so angry because the players are showing no proof of what we are doing. It is not the first time we give a 1-0 lead away for the opponents to score two goals.
“How many times has this happened? When the players don’t learn from this situation then you won’t win games. Maybe the players can give the answers.”
Swansea are now four points off the play-off spots following back-to-back wins, and manager Graham Potter was delighted with his side’s character.
He said: “It’s fantastic for the players and the supporters. It was a pretty awful afternoon in terms of weather. The support was great for us. They stuck with us and helped us.”
Swansea City: Mulder, Roberts, van der Hoorn (Carter-Vickers 46), Rodon, Naughton (Celina 65), Fer, Grimes, Routledge, McKay, Dyer (Montero 63), McBurnie. Unused substitutes: Nordfeldt, Asoro, John, Fulton.
Sheffield Wednesday: Dawson, Lees, Pudil, Thorniley, Baker, Pelupessy, Onomah (Marco Matias 46), Fox (Fletcher 79), Reach, Lucas Joao, Nuhiu. Unused substitutes: Palmer, Jones, van Aken, Wildsmith, Hector.
Referee: D Webb (County Durham).