Gylfi Sigurdsson, Oumar Niasse and Tom Davies netted as the Toffees put last week's 5-1 drubbing behind them to beat the Eagles who, despite Luka Milivojevic's penalty, haven't won a game without Wilfried Zaha now for 18 months.
The result lifts the hosts up to ninth in the standings and leaves Palace just three points above the drop zone.
Sigurdsson drove the game's first real chance wide of the upright just shy of the 10-minute mark after a laboured start to proceedings.
Milivojevic returned the favour not long after but the midfielder saw his drive deflected behind by Idrissa Gueye.
Sigurdsson also saw a free kick from a promising position nodded behind after Gueye had been bundled to the turf by James McArthur, before Niasse was denied a shot on goal by Wayne Hennessey at an acute angle.
17 - Since his debut in January 2012, Gylfi Sigurdsson has scored more goals from outside the box than any other current Premier League player. Ranger.— OptaJoe (@OptaJoe) February 10, 2018
A flashing drive from Gueye was then turned behind by the sprawling Hennessey from a recycled corner while Wayne Rooney slammed a free kick into a Palace wall from a similar position to Sigurdsson's earlier effort.
Davies saw Timothy Fosu-Mensah block his sidefooted effort before Jordan Pickford shovelled Yohan Cabaye's long-range shot behind as the teams traded chances.
Christian Benteke had the ball in the net from the resultant corner but the goal was ruled out for a foul on Pickford, and the Belgian was unlucky to guide a delightful cross from Andros Townsend wide of the post on the stroke of half-time.
It was third time lucky for Sigurdsson just 45 seconds after the restart. Good pressure from Niasse on James Tomkins forced an error and led to the ball dropping for the Iceland international.
A jink to his right to create the space, and Sigurdsson fired home past Hennessey via a deflection off Cabaye to the delight of the Goodison crowd.
Five minutes later and it was 2-0. Cuco Martina picked up a loose ball inside Palace's area, stood up a left foot cross to the six yard line and Eagles pantomime villain Niasse was on hand to nod past Hennessey.
Benteke agonisingly headed wide from Milivojevic's curling centre as a reminder to the hosts that the game wasn't won yet, while Townsend ripped a drive over the bar on the hour.
Roy Hodgson's men continued to press for a goal themselves and Pickford had to be at his best as he tipped Alex Sorloth's header away to safety.
Everton had the points wrapped up 15 minutes from time thanks to Davies' left footed shot.
5 - Luka Milivojevic has converted more penalties than any other player in the Premier League this season (5). Spot.— OptaJoe (@OptaJoe) 10 February 2018
Hennessey was left sprawling by Niasse's deflected effort and, after the Welshman had denied Sigurdsson at a tight angle, the Blues' wonderkid rammed home the rebound.
Palace notched a late consolation after Ashley Williams was deemed to have handled in the area - Milivojevic crashing home from 12 yards - but Sam Allardyce's men had done enough for all three points.