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.
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 opener arrived in the 21st minute. Ashley 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.
After a good spell for Swansea, 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, who nodded home the rebound after Vorm forced his initial effort onto the bar.
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.
Swansea's hopes of a comeback were dealt a terminal blow when half-time substitute 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.
Fellaini wrapped the scoring up with eight minutes remaining as he headed home from Baines' cross with the aid of a deflection off Williams.
Source: PA
Source: PA