Sensational solo goals from Daniel James and Bersant Celina capped off a rousing second-half comeback against Brentford as the Swans booked their place in the FA Cup quarter-finals.
Graham Potter's side had trailed to Ollie Watkins' 28th-minute strike, but they looked a side with far more intent after the break with three goals in an 18-minute spell.
Celina saw his 20-yard free-kick strike the post and then bounce off the back of keeper Luke Daniels and into the net.
James then raced 80 yards to score a stunning individual effort before the Bees were reduced to 10 men after Ezri Konsa hauled down the Wales winger when he latched onto a loose ball and was heading for goal.
It got even better for the hosts as Celina used his trickery to beat two men inside the box before slotting into the far corner.
Connor Roberts thought he had added a fourth after tapping in Celina's parried effort, only for the linesman to raise his flag during the celebrations.
But that mattered little, though, as George Byers added a fourth in the final minute from 20 yards as the Swans booked their place in back-to-back FA Cup quarter-finals for the first time in 92 years.
Oli McBurnie returned to the starting line-up as one of a pair of changes to Graham Potter's side.
The striker - the Swans top scorer this season - has been struggling with illness in recent weeks but was fit enough to feature from the off at the Liberty Stadium.
Kristoffer Nordfeldt started in goal - making his first appearance since the defeat at Rotherham at the start of November - as Erwin Mulder dropped to the bench.
The Swans started brightly, with Byers seeing his cross from a quickly-taken free-kick bounce just wide and evading McBurnie.
But, after that, Brentford were the side on top for the rest of the half.
Following a quick counter attack, Neal Maupay fed Watkins and he fired home past Nordfelfdt from inside the box.
The hosts found it difficult to break down the Bees backline for the rest of the half as Thomas Frank's side were resolute.
But, after the break, the Swans were in full control.
The equaliser came via fortunate circumstances as Celina's 20-yard free-kick struck the post and bounced over the line after deflecting off the back of unfortunate keeper Daniels.
That came three minutes after the break, but there was no fortune about the Swans' second goal.
With 53 minutes on the clock, James blocked a Brentford shot outside his own box and then pinned his ears back and let his pace do the talking. The Wales winger raced more than 80 yards before coolly slotting past Daniels.
He was proving a constant thorn in the side of the Brentford defence, and his pace caused problems once again as Konsa was dismissed for hauling down James when he was racing towards goal after latching onto a loose ball.
That left the Bees having to play the remaining 30 minutes with a man disadvantage, and the Swans took full advantage.
Celina, marked tightly down the left flank, used his wizardry to beat his first marker then outwitted a second challenge before looking up and curling a well-placed effort across Daniels and into the far corner.
Then it was a case of seeing the game out, but Potter's men were not done there as Byers coolly slotted home in the 90th minute after being teed up by Celina.
Swansea City: Kristoffer Nordfeldt; Connor Roberts (Cian Harries, 90), Mike van der Hoorn (captain), Cameron Carter-Vickers, Kyle Naughton; Jay Fulton, Matt Grimes; Daniel James (Joel Asoro, 90), George Byers, Bersant Celina; Oli McBurnie (Courtney Baker-Richardson, 76).
Subs: Erwin Mulder, Luciano Narsingh, Barrie McKay, Yan Dhanda.
Brentford: Luke Daniels, Moses Odubajo, Sergi Canos (Josh Dasilva, 69), Neal Maupay, Ollie Watkins, Kamo Mokotjo (Josh McEachran, 78), Romaine Sawyers, Said Benrahma (Nikolaj Kirk, 69), Julian Jeanvier, Ezri Konsa, Yoann Barbet.
Subs: Dan Bentley, Chiedozie Ogbene, Mads Sorensen, Kolbeinn Finnsson.