David Moyesâ€™ Toffees went to equal top on the table, only behind Chelsea by virtue of goal difference, persisting with their unbeaten league run over the Swans which has now stretched to 15 games.
Swansea were unbeaten in their last five at home but after doing the double last term, Everton made it another victory over the Welsh club thanks to goals from Victor Anichebe, Kevin Mirallas and Marouane Fellaini.
Everton, who ended with 19 shots on target, had applied plenty of pressure on the Swansea defence in the early stages and got their reward in the 22nd minute when Anichebe, starting in place of the injured Nikica Jelavic, fired home from close range for his first back-to-back scoring effort for the club after his goal against Newcastle United on Monday. A long ball into the area was initially well-controlled by Fellainiâ€™s chest before he got a combination of forearm and head to the bouncing ball under duress from Alan Tate and into the path of the Nigerian who volleyed past Michel Vorm with his left foot.
Just as Swansea looked to be getting back into the match, the Toffees struck on the break in fine style when Mirallas nodded home his first Premier League goal. Fellaini played a brilliant ball down the right for Steven Pienaar whose low cross was struck first-time by Mirallas with the original effort knocked onto the underside of the crossbar by Vorm only to see the deflection safely headed into the back of the net by the Belgian.
And it was another Belgian who wrapped up proceedings with nine minutes remaining when Fellaini got on the end of Bainesâ€™ free-kick from the left to get a head on the ball with a little help from the back of Ashley Williamsâ€™ bonce.
The contest was made much more difficult for Michael Laudrupâ€™s outfit when half-time substitute Nathan Dyer was sent off 12 minutes after coming on for a second bookable offence â€“ the first for dissent â€“ before he produced a high boot on Baines leaving referee Anthony Taylor with no choice but to produce a red card.
There were a plethora of chances, first when Mirallas was denied twice by Vorm inside three minutes with a close range shot from Anichebeâ€™s cut-back before his looping header was safely held by the Dutchman.
Swansea improved out of sight after the opening goal and had a great chance to level when Michu rose above the pack to send his header just over the bar from a corner moments before Ki Sung-yeung almost curled one in from outside the area.
In stoppage time of the first half, Everton keeper Tim Howard came up with a vital save to deny Angel Rangel who had been played in by Wayne Routledge one-on-one with the American.
Five minutes into the second half, Howard was at it again this time to keep out Michu who could have laid off for an irate Dyer who was in a much better scoring position.
The Toffees should have been out of sight in the 53rd minute when Mirallas struck the crossbar before Vorm did well to block Anichebeâ€™s follow-up but the visitors then somehow maintained their two-goal lead with Howard to thank â€“ the American getting low to thwart Kiâ€™s powerful effort from the edge of the box.
Fellaini really should have put the game out of reach in the 66th minute when he was found unmarked at the back post by Anichebe but he fluffed his effort wide of the mark before the impressive Anichebe struck a sweet volley on target with ten minutes remaining but Vorm was equal to the task.
Jonathan de Guzman almost grabbed a consolation goal for the Swans only to see his free-kick hit the crossbar before Phil Jagielka courageously put himself on the line to deny Itay Shechterâ€™s effort from the follow-up.
There could have easily been more than three goals scored on the day but in the end the Toffees managed to keep a clean sheet to make three wins and a draw from five matches whilst the Swans slipped down to sixth following their second straight defeat.
Liberty Stadium, Swansea â€“ 20,464