The Toffees dominated from the off and Victor Anichebe fired home from close range after 21 minutes before Kevin Mirallas finished off a superb flowing move just before half-time.
Swansea went close through Angel Rangel and Ki Sung-yueng but their hopes of a comeback were dashed early in the second half when substitute Nathan Dyer was sent off, before Everton's Marouane Fellaini administered the final blow.
Such was Everton's dominance during the first half, it was a surprise it took them so long to open the scoring.
Swansea's sloppiness on the ball and shoddy defending meant goalkeeper Michel Vorm was forced to make three saves inside the first three minutes.
The Dutchman firstly grasped an Anichebe header at the second attempt after the striker had got on the end of Leighton Baines' whipped delivery, before twice denying Mirallas.
The warning was not heeded by the hosts and Ashley Williams endured a few nervous moments when he brought down Anichebe after Wayne Routledge's awful pass had put him in trouble.
The visitors felt the Wales centre-half had been the last man but the covering Alan Tate saved him as he saw yellow as opposed to red.
The opener arrived in the 21st minute. Williams' poor recent run continued as he misjudged a high ball to allow Fellaini to control and tee up Anichebe to fire home from close range, although replays suggested the Belgian had used an arm.
Swansea were guilty of standing off in defence and Anichebe almost punished the struggling Tate when he was given acres of space to turn but fired over.
Home manager Michael Laudrup had been the epitome of frustration during the opening half-hour, but Swansea finally managed to find a spark.
Record signing Pablo Hernandez burst away down the right and his chip was headed off the line by Phil Jagielka after linking with Michu. Phil Neville then blocked Danny Graham's strike after Routledge seized the loose ball.
Ki went closer still for the Swans, clipping a effort inches wide from 25 yards with Tim Howard beaten.
But Everton re-asserted their authority with a wonderful goal.
Neville beat Michu to a long ball near his own right corner, and the former England player's coolness helped start a move that saw Fellaini and Steven Pienaar combine superbly to tee up Mirallas.
Vorm did brilliantly to palm the initial strike onto the bar, but the Belgian was back on his feet quickly to convert the rebound for his first Premier League goal.
Swansea should have trailed by just one goal at the break but Rangel wasted two gilt-edged chances. Howard saved the Spaniard's low shot from Ki's lovely pass, before Rangel inexplicably volleyed away from goal from Routledge's cross.
Laudrup withdrew Hernandez and put on Dyer at the break and the home side increased the pressure as Ki was frustrated by a superb Howard save and the American also denied Michu, while Mirallas struck the bar at the other end.
But Swansea's hopes of a comeback were dealt a terminal blow when Dyer saw red just 12 minutes into the second half. The winger had already been booked for dissent when a clumsy tackle on Baines earned him a second yellow.
Dyer's dismissal sapped the life from what had been an entertaining contest, although Fellaini should have scored only to sky over from Anichebe's ball across the six-yard box.
But the midfielder wrapped the scoring up with eight minutes remaining as he headed home from Baines' cross with the aid of a deflection off Williams.
Swansea's day was summed up when Jonathan de Guzman's superb free-kick was kept out by the post in stoppage time.