They surged into a two-goal lead at the Liberty Stadium thanks to first half strikes from Victor Anichebe and Kevin Mirallas.
Swansea's hopes of avoiding a second successive defeat were in tatters when substitute Nathan Dyer was sent off in the second half.
Marouane Fellaini put the result beyond doubt with a late goal.
"Both the result and performance were very good but the result is the most important thing," said Everton manager David Moyes.
Southampton climbed off the bottom of the table after coming from behind to beat Aston Villa 4-1 at St Mary's.
The Saints, who kicked off without a point after four straight defeats following promotion, must have feared the worst when England striker Darren Bent put Villa ahead in the 36th minute with his first Premier League goal since February.
But Southampton striker Rickie Lambert equalised in the 58th minute when, despite having three defenders around him, he scored with a well-struck shot.
And five minutes later full-back Nathaniel Clyne gave Saints the lead.
Southampton made it 3-1 with a 72nd minute own-goal when Ciaran Clark deflected midfielder Jason Puncheon's shot before Lambert wrapped things up with a last-minute penalty to ease the pressure on manager Nigel Adkins.
"We've been desperate to get some points on the board," said Lambert.
"At half-time Nigel told us certain things were unacceptable but we still believed we could win the game."
Hugo Rodallega scored against his old club as Fulham won 2-1 away to Wigan.
The Colombian striker broke the deadlock when he headed in Dimitar Berbatov's 31st minute cross before Damian Duff's 68th minute goal doubled the lead.
Arouna Kone scored a late goal for Wigan.
Elsewhere, West Bromwich Albion's a 1-0 win away to Reading put them level on points with Everton and left the Royals propping up the Premier League.
The Baggies had to wait until the 71st minute for the decisive goal when a shot from Belgian striker Romelu Lukaku, on loan from Chelsea, found its way through a crowd of players.
"We just kept plugging away," said Baggies boss Steve Clarke. "It is pleasing to look at the stats at the moment but there is a long way to go."
An early goal from Sunderland's 12 million striker Steven Fletcher gave them the lead away to West Ham before Kevin Nolan's 90th-minute equaliser rescued a point for the Hammers in a 1-1 draw.
Sunday sees Liverpool face arch-rivals Manchester United in the first match at Anfield since an independent report cleared the Merseysiders' fans of any responsibility for the deaths of 96 of their fellow supporters during the 1989 Hillsborough disaster.
Meanwhile, champions Manchester City, fresh from their last-minute defeat by Real Madrid, play a rejuvenated Arsenal at Eastlands as Norwich manager Chris Hughton returns to St James' Park for the first time since being sacked by Newcastle.
Tottenham Hotspur, following a goalless Europa League draw with Lazio, welcome London rivals Queens Park Rangers to White Hart Lane.
-AAP