This was all supposed to be about the return of Crystal Palace’s dynamic duo but Everton clearly had not read the script.

They had been pegged back twice after taking the lead thanks to goals from Vitaliy Mykolenko and the hugely impressive Abdoulaye Doucouré, with Eberechi Eze’s virtuoso performance including Palace’s first before Odsonne Édouard finally took one of several chances that fell his way.

Even with Michael Olise making his first appearance of the season off the bench and the hosts sensing blood, it was Everton who maintained their recent superiority against Palace thanks to the substitute Idrissa Gueye’s winner four minutes from time.

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Palace’s dismal record against Everton stretches back to September 2014, having now managed one victory in 18 Premier League meetings since a second successive 3-2 victory at Goodison Park.

Eze’s first start since the end of September in the week that the England midfielder committed his future until 2027 – and Olise’s presence in the match-day squad for the first time this season – provided home supporters with optimism there could be a different outcome this time. But having avoided relegation last season and endured a sticky start to this campaign, Everton have been putting things together of late and they made it four matches unbeaten in all competitions, with six wins in their past nine matches.

“We are showing the belief that we can win these kinds of games and that is really pleasing,” said Sean Dyche. “It’s only the beginning, but we are changing the Everton story.

“There’s a different feel to what people are writing about us and what they are saying and what the team is delivering. It can’t guarantee you wins, but it gives us a much better chance if you’ve got the mentality that we’ve been showing at the moment.”

It took Mykolenko 60 seconds to give the visitors the lead after his shot had been blocked initially and the Palace defence made a hash of clearing the ball. That allowed Jack Harrison to pick out the unmarked left-back for a simple header.

Selhurst Park seemed stunned. Eze, who was left out of the England squad this week as he continues his return from a hamstring injury, had other ideas and within three minutes Palace were level. His mesmerising dribble left several defenders for dead and was ended by a clumsy challenge from Jarrad Branthwaite. Sam Barrott – a 30-year-old taking charge of his fourth Premier League match – had no hesitation in pointing to the spot, with Eze calmly stroking the penalty past the Everton keeper, Jordan Pickford.

Crystal Palace’s Odsonne Édouard scores their second goal.Crystal Palace’s Odsonne Édouard scores their second goal. Photograph: Tony Obrien/Reuters

The frenetic start showed no signs of slowing down as Palace piled on the pressure, Everton almost regained the lead when Dwight McNeil volleyed straight at Sam Johnstone. Eze was shown a yellow card for diving inside the area when he went down after a challenge from Branthwaite that, much to the Palace midfielder’s disappointment, VAR ruled there was not enough conclusive evidence to overturn.

James Tarkowski appeared more comfortable marking Édouard and came to Everton’s rescue when the French striker dithered after being played through by Eze. Having been caught offside on numerous occasions, Édouard finally managed to test Pickford just before half-time but was unable to find a way past.

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Dyche brought on Gueye at the break to replace Amadou Onana and the Senegal midfielder was involved in the move that restored Everton’s lead four minutes into the second half. Mykolenko’s volley from the edge of the area came back off the post and Doucouré reacted quickest to stab the ball past Johnstone, with Jeffrey Schlupp playing him onside.

Once more it was Eze who led the riposte, combining with Joel Ward before bursting into the box and forcing Pickford into a save at his near post. The introduction of Olise was greeted by one of the loudest cheers of the afternoon and he wasted no time in announcing himself with a shot from distance that deflected over the crossbar.

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Roy Hodgson was just about to put Édouard out of his misery by bringing on Jean-Philippe Mateta when Everton failed to clear and Lerma’s header caught Tarkowski napping, with the Palace striker gratefully gobbling up his fifth Premier League goal of the season.

“I thought we did very well to come back from two cold showers at the start of each half,” said Hodgson.

Palace’s hopes of snatching the win were dashed when Gueye played a one-two with Doucouré four minutes from time and poked past a despairing Johnstone to send the travelling supporters into raptures.