The Swans tasted their first defeat in four Barclays Premier League matches as Everton rose into the top four following a 2-1 win at the Liberty.
After a goal-less first half, Roberto Martinez's side went ahead in the 66th minute after Seamus Coleman's stunning strike found the net.
The Swans were on level terms four minutes later as Dwight Tiendalli's volley took a major deflection off Bryan Oviedo and past Tim Howard.
But the Toffees were not to be denied as they regained their lead when Ross Barkley's unstoppable free-kick bounced in off the underside of the crossbar with six minutes to go.
The Swans went in desperate search of an equaliser with Pablo Hernandez and sub Alvaro Vazquez among those to get a chance, but the Toffees - up to fourth and just two points off leaders Liverpool - toasted another victory in SA1.
Defeat meant the Swans dropped to 11th, with 20 points from 17 games.
The Swans made five changes to their starting XI.
Following Michel Vorm's knee operation, Gerhard Tremmel was named in goal while there were two changes in defence as Dwight Tiendalli replaced Angel Rangel - who served a one-match ban - and Chico Flores came in for fellow countryman Jordi Amat.
The other changes from the 1-1 draw at Norwich saw Wayne Routledge start in place of the injured Nathan Dyer while club-record signing Wilfried Bony led the line as Michu missed out through injury.
SWANS:Gerhard Tremmel, Dwight Tiendalli, Chico Flores, Ashley Williams (capt), Ben Davies, Jose Canas, Jonathan de Guzman (Alejandro Pozuelo, 85), Jonjo Shelvey, Wayne Routledge (Roland Lamah, 66), Pablo Hernandez, Wilfried Bony (Alvaro Vazquez, 75).
SUBS: Gregor Zabret, Jordi Amat, Neil Taylor, Leon Britton.
EVERTON: Tim Howard, Phil Jagielka (capt), Bryan Oviedo, Kevin Mirallas (Leon Osman, 78), Sylvain Distin, James McCarthy, Romelu Lukaku, Gareth Barry, Ross Barkley, Steven Pienaar (Steven Naismith, 78), Seamus Coleman.
SUBS: Joel Robles, John Heitinga, John Stones, Apostolos Vellios, Antolin Alcaraz.
The Swans made a bright start, with Jonjo Shelvey and Wayne Routledge involved in the early stages.
First, Shelvey's cross along the face of goal just evaded the winger, before the former Liverpool midfielder then teed up Routledge in the box after some fine footwork, but the latter sliced his effort wide.
Everton looked comfortable on the ball while their pressure game worked well too, and it almost paid off as Kevin Mirallas raced on after the Swans lost possession in their own half, but the winger shot low at Gerhard Tremmel.
Then Ashley Williams came to the rescue in quick succession as he blocked Mirallas' effort after Chico Flores lost possession, and the Swans skipper then picked himself up to snuff out Ross Barkley's cross.
There was another scare for the hosts as a deep Mirallas free-kick was spilled by Tremmel, but the German recovered to take the loose ball as Romelu Lukaku lurked.
The Swans, though, hit back as Routledge cut inside and slipped a through ball for Wilfried Bony to run onto, but the Ivorian blasted his effort high over the crossbar and into the Everton supporters behind the goal.
Bony was at it again before the break, but his attempted overhead kick from Shelvey's flick on failed to connect with enough power.
The Swans probed on two occasions in the final seconds of the half as Jose Canas found himself in space, but he sliced his 25-yard effort wide before Pablo Hernandez fired over from an acute angle
HALF-TIME: Swans 0 Everton 0
Everton started the second half strongly, and while there were few chances, the Toffees create the better of them.
First, Ross Barkley found himself in space in the box, but he stumbled as he was about to shoot, and then Gareth Barry volleyed over after a melee following a corner.
Barkley then went within inches of opening the scoring as he raced on and unleashed a thunderous 20-yard shot which Tremmel did well to tip onto the crossbar and over for a corner
But Everton got their goal soon after, with Seamus Coleman unleashing an unstoppable right-footed effort from 25-yards which curled inside the far post after 66 minutes.
Michael Laudrup immediately made a change, with Roland Lamah on for Routledge.
And the substitute played his part in the equaliser four minutes later.
The Belgian winger touched down for Ben Davies to cross, and Dwight Tiendalli's volley took a major deflection off Bryan Oviedo and past Tim Howard.
And there was another change soon after for the hosts as Alvaro Vazquez replaced Bony with 15 minutes to go.
Everton made a double substitution themselves, with Roberto Martinez introducing Steven Naismith and Leon Osman in place of Steven Pienaar and Mirallas.
Both sides went in search of the game's third goal, and the Swans had a chance as Shelvey slid through a delightful pass the caught the Everton backline flat-footed, but when Alvaro Vazquez raced on down the right, his attempted chip dropped harmlessly into the hands of Howard.
But it was Everton who got the goal.
And it was a piece of magic from Barkley, who lifted his 20-yard free-kick over the wall and into the net via the underside of the crossbar with 84th minute.
Laudrup made his final change as Alejandro Pozuelo came on in place of Jonathan de Guzman as they went in search of an immediate equaliser.
And they almost got it as a Tiendalli cross found its way back to Hernandez, and his powerful drive was punched out by Howard.
And the American had to be at his best in added on time as Lamah's volley from a headed clearance looked goalbound, but the keeper palmed it away.
Then sub Vazquez went close as he attempted to connect with a loose ball in the box, but the Spaniard couldn't get onto the end of it.
That was the end of the action though with time running out on the Swans.