A first half goal from Wilfried Bony and another on the 90th minute from Ashley Williams were not enough for Swansea City as their second trip to Goodison Park this season ended in defeat.
Garry Monk's men were knocked out of the FA Cup at the same venue last month and it was Everton who once again took the win.
Bony's first half goal put the Swans back into contention after Leighton Baines put the home side ahead with a penalty in the 20th minute.
But second half goals from Romelu Lukaku and Ross Barkley sealed the win for Roberto Martinez's team, before Williams secured a consolation goal for the Welsh side.
Swansea City have made three changes to their starting line-up from the one which lost to West Brom last weekend.
Chico, Dyer and Hernandez come in for Amat, De Guzman and Lamah.
It's the Swans' second trip to Goodison Park this season, after they were knocked out of the FA Cup at the same venue last month.
Swansea City will be looking for points against Everton today, after a home defeat to fellow strugglers West Brom last weekend pulled them deeper into relegation trouble.
It's the Swans' second trip to Goodison Park this season, after they were knocked out of the FA Cup in a 3-1 defeat last month.
Boss Gary Monk says he'll be taking an almost fully fit squad, which could be a much needed boost to their trip.
Swansea are looking nervously over their shoulders. At the moment, they're ahead of 18th placed Sunderland but their cup exploits mean they have three matches in hand.
The form of the Swans doesn’t offer any confidence for fans with two wins in 15 Premier League games – and just one in five since Garry Monk was appointed interim manager in February.
Ross Barkley enhanced his ever-growing reputation by scoring from a late free-kick to give Everton a 2-1 win against Swansea City at the Liberty Stadium.
A fierce strike from Seamus Coleman had given the visitors a 66th-minute lead with a ferocious strike, but they were pegged back when Bryan Oviedo put Dwight Tiendalli's off-target effort into his own net.
But Everton were not to be denied by the Swans as midfielder Barkley stepped up with six minutes to go to beat Gerhard Tremmel with a strike which nicked the underside of the bar on its way in.