The hosts enjoyed some good possession early on but were unable to break the deadlock, and they were ultimately punished by some clinical finishing from the visitors
The home side dominated the early stages of the game yet failed to create any clear cut chances, with the Everton back line holding strong.
As the game wore on it was the visitors who took control, and after Leighton Baines had opened the scoring with a scintillating free-kick, David Moyes’ men never looked back, with Nikica Jelavic securing the points late on.
A sun drenched Liberty Stadium was treated to an equally bright start from the home side, with Danny Graham sent through on goal by Leon Britton early on.
The Swans’ top scorer composed himself before firing straight at Tim Howard, when Scott Sinclair was perhaps better placed had Graham looked up.
The home side enjoyed the lion’s share of possession throughout the first half, yet was unable to fashion any clear cut chances, with Sylvain Distin and Jagielka producing a commanding defensive display.
However, as the half began to draw to a close it was the visitors who looked the more likely of the two sides, but the Toffees were unable to utilize the height of Jelavic and Distin from set pieces.
Both sides emerged unchanged for the second half, and the half began as the first had ended, with the visitors on top.
And they perhaps should have taken the lead just a minute in, when Swansea captain Williams was caught in possession on the edge of his own box. The ball broke for Jelavic who had to stretch before firing over Michel Vorm’s crossbar.
After Gibson had forced Vorm into action with a stunning left-footed drive, the ball broke for Jelavic who improvised well to hook the ball back in the danger area, but the cross narrowly eluded the advancing Tim Cahill.
Yet the visitors didn’t have to wait long to make their dominance count.
Swansea defender Williams conceded a cheap free-kick, hauling Baines to the ground on the edge of the area. Jelavic and Baines stood over the free-kick, but it was the left-back who stepped up to curl a sublime effort over the wall and into the back of Vorm’s net, leaving the Dutchman grasping at thin air.
With Swansea attacks few and far between, Rogers decided to roll the dice late on, introducing Josh McEachran, Leroy Lita and Luke Moore.
But the changes had little impact on the flow of the game, and the visitors should have put the game to bed just minutes later.
Steven Pienaar and Jelavic carved the Swansea back line open, leaving the Croatia international one on one with Vorm, yet the forward somehow contrived to slot the ball the wrong side of the post.
However, the former Rangers man didn’t have to wait long to redeem himself.
After substitute Fellaini had outmuscled Williams on the Swansea byline, the Belgian cut the ball back to Jelavic who was left with a simple tap in to record his second goal for the club.
The defeat is just Swansea’s third at home all season, and means Everton leapfrogs the Swans into ninth, while Swansea drops two places to tenth.