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 compose 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 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 the Owls manager.
“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 playoffs following back-to-back wins, and Graham Potter was delighted with the character his side showed.
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. I think we started off well in difficult conditions but, as in anything, the opposition came back into it a little bit.
“Overall, in that type of afternoon there’s going to be a couple of mistakes. Maybe there was a little bit of that in the first goal and it got nervy for a few minutes but I thought overall we deserved the victory.
“It was great for Bersant to come on and score and turn the game around for us. As a team, it shows there’s a mentality to recover from setbacks, recover from mistakes.
“In the end, I’m really proud of the players’ effort, togetherness and character, and also really proud of the supporters because it was an awful day but they were fantastic.”