David Moyes’s European dream is now hanging by a slender thread. Against a Crystal Palace side who have been otherwise distracted by their Conference League exploits, Everton were unable to take their opportunity to close the gap on their rivals as Jean-Philippe Mateta came off the bench to deny them victory.

It could have been even worse for Moyes had the referee, Tom Bramall, awarded a penalty when James Garner hauled down Jørgen Strand Larsen just before half-time or Adam Wharton’s shot not struck the outside of the post in the 90th minute.

A frantic finish that saw Dean Henderson deny Iliman Ndiaye in stoppage time before Mateta missed a great chance to win it for Oliver Glasner’s side meant Everton stretched their unbeaten run against Palace to 11 matches.

But it means they must now rely on other teams dropping points if they are to have a chance of qualifying for Europe next season, while Palace are still not quite mathematically safe from relegation just yet.

Their last Premier League win against Everton came in December 2021 when Rafael Benitez was in charge of the visitors and the smart money seemed to be on that run continuing given their opponents’ European exertions. Four of Palace’s players have already played more than 4,000 minutes this season, with Maxence Lacroix starting his 52nd game of a marathon season that could end with him being called up for France’s World Cup squad despite being culpable for Beto’s goal.

But in a move that suggested he will rotate more when they travel to Manchester City on Wednesday, Glasner decided to select virtually a full-strength side with his record signings Brennan Johnson and Strand Larsen the only two changes from the win over Shakhtar Donetsk.

Everton fielded the same team from their thrilling draw against City that saw them set an unwanted record of becoming the first team in the Premier League to concede injury-time goals in three successive matches. Moyes had lamented those missed opportunities but knew their chances of qualifying for Europe next season would be greatly improved with a victory here.

It didn’t take long for Everton’s players to show they were up for the challenge. Palace’s achilles heel this season has been an inability to defend from set pieces and there was no excuse for James Tarkowski being unmarked at the back post when Kiernan Dewsbury-Hall’s ball deflected to him. It was the 18th goal Palace have conceded from a set piece in the Premier League – more than any other team.

Chris Richards almost made amends at the other end when he headed over from a corner before Jordan Pickford, up against his England rival Henderson and being watched by Thomas Tuchel in the stands, pulled off an excellent save to keep out Lacroix’s shot. Ndiaye broke downfield but couldn’t take the chance to double Everton’s lead. The Senegal forward was made to regret that when his compatriot Ismaïla Sarr slammed home after his initial shot was blocked by Michael Keane. What a season it has been for him.

A few words were exchanged between the benches after Tarkowski went unpunished for taking out Sarr off the ball before James Garner escaped what looked like a clear push on Strand Larsen inside the area. It was Henderson’s turn to rescue Palace just before half-time when another Everton corner caused havoc and Sarr was on hand to clear.

Less than two minutes of the second half had elapsed when Everton regained their lead. Tarkowski only intended to clear the ball from his own half but it looped perfectly into the path of Beto, who was able to turn inside Lacroix before beating Henderson.

The Palace goalkeeper had to be at full stretch to tip over a free-kick from Garner. The introduction of Mateta from the bench with 25 minutes to play gave Palace more impetus as they strived for an equaliser. Sarr was inches away from diverting Wharton’s floated ball past Pickford for his second of the afternoon before Tyrick Mitchell’s clever cross teed up the France striker to set up a grandstand finish that left Everton wondering what might have been.