The Toffees' Vitaliy Mykolenko scored in the first minute, then reclaimed the lead through Abdoulaye Doucoure just after halftime. Doucoure then set up Idrissa Gana Gueye in the 86th minute for a match-winner in South London.
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Everton, meanwhile, is now unbeaten in three Premier League matches, the seven gained points boosting them into 14th with 14 points. That's one behind Palace and eight clear of the bottom three.
Palace was coming off a 2-0 win at Burnley that moved the Eagles onto 15 points a tie for 10th place on the table after 11 games. They've slipped into 12th.
Everton win the type that’s the product of belief, can be the source of more
First of all, Everton supporter, we understand how difficult it is to believe positive things about your beleaguered club, but let's be real; All of the recent signs say Sean Dyche has your Toffees operating like a real, working football club.
This is a mid-table team with top-half potential, sitting inside the top 10 in expected goals, created chances, and a number of defensive metrics.
They still aren't possessing the ball but they'll foul the heck out of you when there's danger in sight, and now they are showing more and more belief in what Dyche is preaching will lead to success.
A first-minute away lead blown within five minutes might've spelled doom for any number of iterations of Everton, but this team reclaimed the lead twice on the day. The Toffees are unlikely to be part of the relegation fight this season. That that matters speaks volumes to how far the club had fallen, but there's real relief there. And when you're not in a relegation race, you can build each week instead of worrying how badly a failure will set you back.
Palace getting healthy and — mercifully — less boring
Competition for attacking places on this team will drive the Eagles forward. Odsonne Edouard's been good but Jean-Philippe Mateta has proven capable. Olise, Eze, Jordan Ayew, and Matheus Franca all can't play if either Edouard or Mateta is out there. That's not even mentioning Jeff Schlupp.
The defending was less than adequate on Saturday, but Hodgson can keep that settled especially with his midfield pieces ready to fight. Palace, while healthy, will no longer be a slog for the eyes. That's a win for the neutral and for the Eagles fan base.
Palace goes to Luton Town in two weeks, a 10am ET Saturday kickoff after the international break.
Everton hosts Manchester United at 11:30am ET Sunday (November 26).
Crystal Palace vs Everton final score 2-3
Goals: Mykolenko (1'), Eze (5'), Doucoure (50'), Edouard (73'), Gueye (86')
Crystal Palace vs Everton player ratings
Idrissa Gana Gueye goal: Clever combination with Doucoure
What a game this has been!
Idrissa Gana Gueye has the ball on the right on an incisive pass from Tarkowski, he carries and squares it inside for Doucoure.
His run doesn't stop, and Doucoure sends him into the box. A slightly heavy touch means Gueye has to slide to shoot, and he cuts his effort inside the near post. 3-2!
Odsonne Edouard goal: Palace levels it
There's a long VAR review to decide whether Odsonne Edouard has his sixth goal of the Premier League season (on his sixth shot of the night).
The offside reviews do not deny the ex-Celtic star his latest marker, as he's so technical in taking the ball out of the air and beating Jordan Pickford.
Abdoulaye Doucoure goal: Rebound fun
Everton gets a goal off a bit of luck, a bit of technique, and a bit of hope.
It's luck that sees a shanked cross head for Mykolenko.
It's marvelous technique that the Ukranian smashes on goal.
It's the hope of a rebound that allows a just-onside Doucoure to snap home for his fourth Premier League goal of the season.
Crystal Palace vs Everton halftime analysis: 1-1
That was a really good watch for a neutral.
Partisan folks will think that either Palace or Everton should've gotten more fouls called against them, and Eberechi Eze will feel wronged by a yellow card for simulation.
Both teams have taken four shots and committed five fouls. Palace have been slightly better but that's down to Everton's plan to possess when possible but be content to counter attack.
James Garner and James Tarkowski have starred for Everton, while Palace's Eze is the star of the game. Essentially, he's been "Mr. Flashpoint," too.
Eberechi Eze goal: Palace star converts penalty
Mark this one down as a penalty won.
Eberechi Eze darts into the box and feasts on a stretched leg from Jarrod Branthwaite.
VAR won't see enough to overturn it, and Eze's stuttered run-up fools Jordan Pickford. 1-1!
Vitaliy Mykolenko goal: Toffees score inside 60 seconds
Well, here's a start!
Jack Harrison stands up a cross to the heart of the six and Mykolenko gets high to nod home.
What a start for Sean Dyche!
How to watch Crystal Palace vs Everton live, stream link and start time
Kickoff: 10am ET Saturday
TV Channel: Peacock Premium
Stream: Premier League on Peacock
Crystal Palace lineup
Focus on Crystal Palace
Eberechi Eze's return has been welcomed, and Michael Olise's coming back soon to join Odsonne Edouard, Matheus Franca, Jean-Philippe Mateta, and Jordan Ayew in a much deeper attack.
Focus on Everton
Dominic Calvert-Lewin is getting back to his best, and improvement can be found if Jack Harrison and Co. stay healthy and deliver more action to the danger areas.