Leighton Baines, Romelu Lukaku and Ross Barkley were on target to secure a 11th home league win for Roberto Martinez's men this season, moving them above Tottenham, who host Southampton on Sunday.
Baines opened the scoring with a penalty, maintaining his 100 per cent record from the spot in the Premier League, only for Wilfried Bony to pull Swansea level before half-time.
Lukaku and Barkley then found the net early in the second half to take the game away from the visitors and leave Garry Monk still searching for his second win since succeeding Michael Laudrup.
Ashley Williams pulled a goal back in stoppage time, but it was not enough for Swansea, who will be looking nervously over their shoulders towards the relegation zone.
The hosts began well and could have opened the scoring as early as the seventh minute.
Seamus Coleman played a neat one-two with Lukaku and rolled a cross to Barkley, but the midfielder skewed his effort wide of goal.
Baines made sure Everton did not have to wait long to move ahead as he emphatically converted a penalty following Chico Flores' foul on Barkley after 19 minutes.
John Stones released Barkley with a cross-field pass out of defence, and the 20-year-old headed straight for goal before being clipped by the Spanish defender just inside the area.
Swansea drew level on 32 minutes after Wayne Routledge showed fantastic vision to find Angel Rangel on the right of the penalty area. Rangel fired a low cross into the six-yard box and Bony was there to tap home.
Pablo Hernandez then went close to giving Swansea the lead two minutes before half-time, with a free-kick that took a deflection to wrong-foot Tim Howard but rolled narrowly wide.
After Sylvain Distin had missed a chance to put Everton back in front early in the second half, the hosts soon restored their lead as Lukaku fired home on 53 minutes.
The Belgium international found compatriot Kevin Mirallas, who played an inviting cross into the six-yard box for Lukaku to pounce at the near post.
Barkley made it 3-1 five minutes later following some poor defending from Swansea.
Mirallas was again the architect as his corner enabled the unmarked Barkley to convert a simple header.
Bony had a chance to narrow Swansea's deficit just after the hour mark, but the Ivory Coast international saw his header palmed away by Howard, before Coleman completed the clearance.
Howard was again on hand to deny Hernandez in the closing stages, but there was nothing he could do to stop Williams' injury-time header from a right-wing corner.
