Crystal Palace and Everton finished all square 0-0 at Selhurst Park on Saturday.
A seventh-placed finish could prove enough to play Europa League football next season but, after thrashing Manchester United last weekend, the Toffees could not find a way past Palace.
With fellow European hopefuls Wolves winning at Watford, Everton are now four points adrift of Nuno Espirito Santo's side.
Everton boss Marco Silva said, "We deserved the three points.
"It was clear we were the best team on the pitch and we tried everything to win the match.
"Two shots off the goal, a fantastic save from the goalkeeper and from the first minute we came here to dominate the game and you know how difficult it is to do that here.
"Being more clinical this afternoon we win this game clearly.
"We started the game with that chance for Dominic (Calvert-Lewin) and after that we had three big, big chances to score. We didn't score but our players tried everything and deserved more."