The Imps led at half-time through Reeco Hackett's penalty but goals after the break from Moussa Yeo and Stephen Eustaquio turned the tide the way of the Welsh side.
"It's a nice feeling, of course, but we need to stay grounded as we know there are a lot of games and there is a lot still to compete for," said Swansea boss Vitor Matos, with three weeks until his side's next fixture.
"We have an international break now, so we will not feel the effect of this game on the next game because it is so far away."
West Ham striker Taty Castellanos was accused of spitting at Millwall goalkeeper Filip Marschall during a fiery 2-2 London derby draw at the Den.
Marschall lodged a complaint with the match officials after the incident in the 65th minute of the clash.
With West Ham trailing 2-0, Argentinian Castellanos came to close down Marschall as he held on to the ball.
The players clashed and Marschall, on his Championship debut, then appeared to point at his shirt, gesturing he had been spat at.
"I haven't asked him about it, but it's like anything else, I'm sure there'll be footage of it, or not of it if it didn't happen," said Millwall manager Alex Neil.
Castellanos could face a six-match ban if he is found to have spat at Marschall, which would be a huge blow to West Ham's promotion bid.
Hammers boss Nuno Espirito Santo said: "I'm aware of the allegation, I didn't see the images, I didn't speak with Taty."
The controversy was a sour twist to a first meeting between the fierce London rivals in nearly 15 years.
Millwall led at half-time through goals from Caleb Taylor and Camiel Neghli.
But in a vastly improved second-half display the visitors hit back through Mohamadou Kante and Konstantinos Mavropanos.
Wrexham bounced back from a 6-0 thrashing by West Ham with a 2-1 win over Southampton in another fixture which featured the after-effects of the Saints' Spygate scandal of last season.
Welsh side Wrexham ended the 2025/26 campaign in seventh place but held hopes of being included in the play-offs after Southampton were expelled from the final for spying on opposition training sessions in breach of English Football League rules.
Ahead of Saturday's match, Wrexham's Hollywood co-owners Ryan Reynolds and Rob Mac had a playful dig at Southampton in a 59-second video posted on social media, while home fans chanted "you only win when you're spying" during the game.
England's governing Football Association will belatedly hold a hearing into Southampton manager Tonda Eckert's role in the affair next week and any disciplinary action could see the German miss the south coast derby against Portsmouth.
Eckert, asked if he was expecting a touchline ban, said: "Let's see. I think it's been a situation that has dragged on for a long, long time and I think we will have some clarity very soon."