Nathan Dyer was sent off in the defeat
Michael Laudrup's side lost 3-0 at the Liberty Stadium
The loss is the Swans' second in a row, and sees them slip to 9th in the Premier League.
Victor Anichebe opened the scoring in controversial fashion for the visitors when he volleyed the ball into the net after a suspected handball from Marouane Fellaini.
Swansea City began to come into the game after the half hour mark, coming close to levelling the scores with a number of chances through star signing Michu and Angel Rangel.
Just as the Swans looked as though they may equalise before the break, Belgian forward Kevin Mirallas broke free to double Everton's lead.
After the break, Swansea were reduced to ten men when Nathan Dyer, brought on as a replacement for Pablo Hernandez, picked up a second yellow card for a challenge on Leighton Baines.
Everton took full advantage of the extra man, with Marouane Fellaini heading home with just under ten minutes left on the clock to wrap up the three points for the visitors.
Speaking after the game, Swans boss Michael Laudrup said:
"I have to say, Everton deserved the win".
"They were better than us".
"The first 30 minutes laid the ground work for the defeat, we gave away a lot of fouls and played too slow".
"We are still a good team - it is never good to lose, especially not in this way, but I think this result could be a good wake up call for everybody at the club".
The Swans will look to get back to winning ways in the Premier League next weekend, when they travel to the Potteries to face Stoke City.
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