The Swans claimed a historic win in the FA Cup after Wilfried Bony's 90th minute winner sent Manchester United out in the third round at Old Trafford.
Wayne Routledge had given the Swans a shock 12th minute lead after he lobbed Anders Lindegaard following Alejandro Pozuelo's superb through ball.
But United would equalise four minutes later via Javier Hernandez, who volleyed home from Alexander Buttner's cross.
In the second half, the Swans held firm while United were reduced to ten men with just ten minutes left as sub Fabio - on for injured skipper Rio Ferdinand - was shown a straight red card following on fellow replacement Jose Canas.
Then came Bony's brilliance in the final minute of normal time as he rose highest at the near post to power home Routledge's perfect cross to claim the Swans' first ever win at Old Trafford.
It means the Swans will now face either Birmingham, Crawley Town or Bristol Rovers - the latter two face a second-round replay - on January 25/46 in the fourth round.
The Swans made five changes to the starting XI.
Three changes were made in defence with Dwight Tiendalli and Neil Taylor the starting full-backs while Jordi Amat came in to partner fellow Spaniard Chico Flores.
In midfield, Leon Britton returned from illness to skipper the side while Alejandro Pozuelo started on the wing in place of Roland Lamah, who had picked up a groin injury.
Also unavailable through injury were Michel Vorm, Nathan Dyer, Pablo Hernandez and Michu.
MAN UTD: Anders Lindegaard, Rio Ferdinand (capt,Fabio Da Silva, 76), Jonny Evans, Chris Smalling, Javier Hernandez, Danny Welbeck, Tom Cleverley, Darren Fletcher, Antonio Valencia (Adnan Januzaj, 63), Shinji Kagawa, Alexander Buttner.
SUBS: David De Gea, Anderson, Ryan Giggs, Michael Carrick, Wilfried Zaha.
SWANS: Gerhard Tremmel, Dwight Tiendalli, Chico Flores, Jordi Amat, Neil Taylor, Leon Britton (capt), Jonathan de Guzman (Alvaro Vazquez, 85), Jonjo Shelvey (Jose Canas, 64), Alejandro Pozuelo, Wayne Routledge, Wilfried Bony.
SUBS: David Cornell, Angel Rangel, Ashley Williams, Ben Davies, Rory Donnelly.
In what was a promising start to the game, the hosts went close after a neat one-touch move involving Danny Welbeck and Shinji Kagawa eventually teed up Javier Hernandez, but the Mexican failed to make clean contact with his attempt from just 10 yards out.
Soon after the Swans supporters were on their feet as Chico Flores and Wilfried Bony linked up to find Alejandro Pozuelo, and the Spaniard's sublime first-touch pass sent Wayne Routledge in on goal, and the winger's perfectly judged chip dropped over Anders Lindegaard and into the net after 12 minutes.
The lead, however, wouldn't last long, though, as just four minutes later Hernandez made up for his earlier miss by volleying home Alexander Buttner's pinpoint cross from the left flank.
It could have got worse for the Swans not long after the restart as Welbeck teed up Antonio Valencia, but the winger's effort raised over the crossbar before Chico Flores came to the rescue with a sliding tackle to deny Kagawa when he was put in on goal.
The Swans replied with a Jonathan de Guzman free-kick, which Lindegaard dealt with comfortably, while Tremmel then had to gather Tom Cleverley's low free-kick at the other end.
Manchester United would enjoy the better possession and territory for the rest of the half, with Hernandez volleying wide from another excellent cross from the left flank.
But the Swans went into the break on level terms.
HALF-TIME: Manchester United 1 Swans 1
The Swans made an encouraging start to the second half as Routledge saw his shot deflected over after his intended pass to Pozuelo found its way back to him.
They continued to prove, and Pozuelo dragged his effort off target after a neat run into the United box before Bony held off Cleverley and then raced forward 20 yards but his shot squirmed wide.
Just past the hour mark, Michael Laudrup made a change as Jose Canas replaced Jonjo Shelvey.
The hosts were also forced into a change as Fabio took the place of the injured Rio Ferdinand in the 76th minute.
But the defender would only last four minutes, as his sliding challenge on Canas was deemed high and dangerous by referee Mike Dean, and he showed him the red card.
And the Swans would take full advantage.
With 90 minutes on the clock, Routledge beat his marker Darren Fletcher down the left before his cross found Bony, who powered his header past Lindegaard from six yards.
The Swans kept pushing for a third in added on time, and Bony again went close after being teed up by Pozuelo, and then sub Alvaro Vazquez curled just wide from 20 yards.
But the final whistle blew soon after, with the Swans claiming a famous cup victory at Old Trafford.