Everton romped to victory at the Liberty as the Swans suffered a second consecutive Premier League defeat - their biggest on home soil since August 2009.
Victor Anichebe put the the visitors ahead after 21 minutes when he drilled home low down despite Marouane Fellaini appearing to use a hand in the build-up.
David Moyes's side were 2-0 just before the break as Kevin Mirallas nodded home from close range after Michel Vorm had directed his shot onto the crossbar.
The Swans had gone close on a number of occasions through Angel Rangel, Ki Sung-Yueng and Pablo Hernandez, but they couldn't find a way through.
Substitute Nathan Dyer, introduced at the break, then saw red just 12 minutes after coming on as he received two bookings.
The man advantage put the Swans under extra strain, and Fellaini headed past Vorm following a deflection off Ashley Williams to put the icing on a deserved win.
It wasn't the Swans's afternoon, with Jonathan de Guzman hitting the crossbar with a late free-kick, and then Wayne Routledge's shot was well blocked by a solid Everton defence.
It was the heaviest home loss for the Swans since Middlesbrough claimed a 3-0 victory in SA1 just over two years ago.
The Everton hoodoo continues - the Swans never having beating the Toffees - with attention now turning to Tuesday's Capital One Cup tie at Crawley.
Michael Laudrup handed home Premier League debuts to club record signing Pablo Hernandez and Ki Sung-Yueng.
Hernandez, who came off the bench at Aston Villa last time out, came in to replace Nathan Dyer, who dropped to the bench. Ki, whose only appearance so far has come in the Capital One Cup, got his first taste of the top flight with Leon Britton dropping to the bench.
Garry Monk was among the replacements along with Dwight Tiendalli, who made the matchday squad for the first time since joining on a free transfer.
The Swans were without Neil Taylor, who is out for the rest of the season, and the suspended Chico Flores.
Everton boss David Moyes, meanwhile, was without the injured trio of Nikica Jelavic, Darron Gibson and Tony Hibbert.
Michel Vorm, Angel Rangel, Alan Tate
(Leon Britton, 55), Ashley Williams (capt), Ben Davies, Ki Sung-Yueng, Jonathan de Guzman, Michu, Pablo Hernandez (Nathan Dyer, ht), Wayne Routledge, Danny Graham (Itay Shechter, 72).
Gerhard Tremmel, Dwight Tiendalli, Garry Monk, Luke Moore.
Tom Howardl, Leighton Baines, John Heitinga, Phil Jagielka, Kevin Mirallas (Steven Naismith, 71), Phil Neville (capt), Leon Osman, Steven Pienaar (Magaye Gueye, 86), Seamus Coleman, Marouane Fellaini (Bryan Oviedo, 86), Victor Anichebe.
Jan Mucha, Sylvain Distin, Apostolos Vellios, Shane Duffy.
Before kick-off, the Robbie James Wall of Fame was unveiled.
The first 20 inductees inscribed onto the honour roll were Jack Fowler, Roger Freestone, Bob Latchford, Mel Nurse, Herbie Williams, Alan Curtis, Cyril Pearce, Harry Griffiths, Ivor Allchurch, James Thomas, Nick Cusack, Brian Evans, Keith Walker, John Toshack, Wilf Milne, Lee Trundle, Wyndham Evans, Dudley Lewis, Len Thompson, John Cornforth.
The stadium then paid tribute to former chairman Malcolm Struel, who had passed away in the week, while there was also remembrance for the victims of the Hillsborough disaster following on from the findings of the latest report into the tragedy.
Then it was game-time, and Everton were quickly into their stride, a Leighton Baines free-kick was nodded goalwards by Victor Anichebe, but Michel Vorm gathered after his initial parry with 70 seconds on the clock.
The action continued as after a scramble following a blocked Victor Anichebe cross, Kevin Mirallas reacted first to the loose but his header failed to trouble Vorm.
Then it was the Swans' time to attack, Pablo Hernandez found himself in space down the right but when he tried to find countryman Michu, his pass was off course.
The first big call of the game referee Anthony Taylor adjudged Ashley Williams to have taken down Anichebe 20 yards out from goal as the striker latched onto a Wayne Routledge mis-pass. Williams was booked, with the away side feeling that the Welsh international was the last man.
Back at the other end, Hernandez volleyed wide after Danny Graham had done well to hold off John Heitinga and chest the ball down.
Anichebe used his strength well to buy himself some space before shooting on the turn straight at Vorm, before the striker did get his goal.
A Baines free-kick from just inside the Swans half was brought down by Marouane Fellaini, and when the loose fell to Anichebe he drilled low into the corner of the net after 21 minutes.
The Swans then came alive. Ashley Williams' superb crossfield ball to Angel Rangel was headed on to Hernandez, who also beat his marker to the header and raced on. He touched on to Michu, who rounded Tim Howard at a tight angle and fed back to Hernandez, but his goalbound chipped effort was nodded off the line.
Then it was Routledge's turn to test the Everton backline, and when Michu slipped through a neat pass inside the box, it took a superbly-timed block from Phil Jagielka to nudged it wide for a corner. From the delivery, Michu rose highest but his header flew just over the crossbar.
The home side were applying more pressure, and Ki Sung-Yueng went within inches of finding the top corner with a superb drive, but Howard breathed a huge sigh of relief as he watched the shot drop wide.
But the visitors hit back with a fine counter to double their lead in the 44 minute.
After Fellaini sent an excellent through ball down Everton's right flank, Pienaar's cross was excellently placed for Mirallas. Vorm touched his first effort onto the crossbar, but the striker was at hand to dive in and head over the line from two yards.
Two minutes of added on time were to be played, and Graham went close to pulling one back but he failed to connect with Rangel's cross. Then the Spaniard went close on two occasions.
First, he raced onto Ki's through ball in the box but saw his effort cannon off the underneath of Howard before bouncing over for a corner.
Then he snuck in at the back post to meet Routledge's cross, but the right-back sliced his effort back in play rather than towards goal.
Half-time: Swans 0 Everton 2
Laudrup made a change at the break, Nathan Dyer replacing Hernandez as the Swans looked to get back into the contest immediately.
And the Swans did make an encouraging start. After Howard cleared straight to Ki, the South Korean found Michu in space. He raced on towards goal, but Howard deflected his shot wide for a corner after 49 minutes.
The game exploded into life once again, with Fellaini latching onto Anichebe's through ball, and after some heroics by Ben Davies, Mirallas gathered loose ball before his effort cannoned out off the crossbar.
The ball quickly headed up the other end and, after a neat passage of play, Ki almost reduced the deficit but his shot again hit the underside of Howard before bouncing over the crossbar for a corner.
Leon Britton then came on in place of Alan Tate
, but, with 57 minutes on the clock, Dyer was dismissed after collecting his second booking for foul on Baines.
It meant a testing 30 minutes for the Swans, who had now dropped Ki into defence.
Fellaini fired over from eight yards from Anichebe's cross before both managers made a change with 20 minutes to go.
Itay Shechter replaced Graham while for the visitors Steven Naismith came on for scorer Mirallas.
Everton went within inches of making it 3-0 as Fellaini met Baines' corner, but Britton was there to head off the line.
Anichebe then volleyed wide from another Baines cross after another swift Everton counter, before the powerful striker saw another volley parried low down by Vorm. The loose bounced back to Anichebe, and his resulting cross thundered into the sidenetting.
Everton were not to be denied any longer. A Baines free-kick found Fellaini, and the big Belgian's header deflected off Williams and into the net.
The Swans pushed hard for a consolation, and Jonathan de Guzman beat Howard but not the crossbar as his free-kick cannoned off the woodwork.
It wasn't to be for the Swans, with Everton inflicting a second consecutive defeat on Laudrup's side.