James Tarkowski headed Everton in front from an early corner and Beto restored the lead after Ismaila Sarr's equaliser, before Crystal Palace's Jean-Philippe Mateta fired in Tyrick Mitchell's cutback with 14 minutes remaining to deny Moyes a first Everton win since March 21.
The sequence makes Everton the first side in Premier League history to concede a result-altering goal in the 90th minute or later in three consecutive matches — having previously been denied victories against Liverpool and West Ham in similar fashion.
Speaking to the Liverpool Echo afterwards, Moyes did not disguise his frustration.
"Good sides have to do the business at this time of the season. If you really want to be in it then you've got to get wins and that's why we see the big teams there. We have got to learn to deal with the pressure of winning the games and how we can do it."
He acknowledged the facts plainly.
"The facts would also say that we've probably not been able to handle bits of it. And we have to try and do that better — but we've not been in this position for a long time."
On the European push specifically, he remained cautiously optimistic while accepting the reality.
"We're still not out of it yet, really. So I'm hoping that we can somehow muster something up. But if not, we'll have to accept it and move on. It's not gone yet but it's drifting away from us at the moment."
Everton have not played in European football since 2017/18. The prospect of ending that wait had been genuine heading into the final weeks. Moyes, who ended West Ham's 43-year trophy drought by winning the Europa Conference League in 2023, had spoken openly about "dreaming" of delivering the same for Everton supporters.
Two matches remain against Leicester City and Brentford. Seventh place is the likely European threshold. The gap requires results elsewhere to fall Everton's way — and a side that has lost two points per game in the closing run will need to find something different quickly.