Crystal Palace substitute Michy Batshuayi ensured it was another frustrating night for Carlo Ancelotti's men at Goodison Park, as Everton failed to resurrect their hopes of securing European football for next season.
James Rodriguez fired the hosts ahead just before the hour mark, but Everton were unable to capitalise on their chances as Batshuayi rescued a deserved point for the away side in the dying minuets, to scupper The Toffees' quest for European football.
Story of the match
Ancelotti made four changes to the side that were defeated by Manchester City in the FA Cup quarter-final last time out. Robin Olsen, Michael Keane, Tom Davies and Rodriguez returned to the XI at the expense of Joao Virginia, Ben Godfrey, Gylfi Sigurdsson and Allan.
It was the away side who had the better start to the fixture, with Everton displaying symptoms of a post-international break hangover. Palace, who lined up with their creative trio of Eberechi Eze, Jordan Ayew and Wilfred Zaha behind Christian Benteke, showed early signs of causing an upset on Merseyside.
The Eagles' first chance of the game saw Benteke and Zaha link-up and get in behind the home sides defence, with Zaha's subsequent shot deflecting behind for a corner kick.
Everton sprung into life from the resulting corner as they unleashed a counter attack spearheaded by Andre Gomes as he pounced on the dawdling by Palace skipper Luka Milivojevic, allowing Gomes to send Richarlison away. The Brazilian attempted to slide the ball across to strike partner Dominic Calvert-Lewin, however, he was flagged offside for his role in receiving the initial ball from Gomes.
You'd be forgiven for thinking that Hodgson's side were the ones at home, as they dominated possession of the ball early on. Everton appeared to be lining up in a 5-3-2 as The Toffees seemed happy to sacrifice their share of the ball in the hope of following the blueprint of so many who have been victorious against them at Goodison Park this season, with quick turnovers and clinical finishing.
Benteke continued to be the provider of all Palace chances as Zaha and co used his presence to bounce the ball off, in return of a delicate pass through the Everton backline.
Calvert-Lewin should have opened the scoring with his 15th goal of the season after an eye of the needle pass from Mason Holgate sent him through with just Vincente Guaita to beat, but his eventual effort caused little trouble for the Palace 'keeper.
Everton found themselves growing into the match as they seemed to shake off any potential sluggishness from the international break. Seamus Coleman was the next to attempt to unlock the stern South-London defence just after the half-way mark of the first half. He waltzed past Patrick van Aanholt with ease, before his floated cross towards the back post was unable to be finished off by the head of Richarlison.
The Brazilian was presented with the best chance of the half shortly after his earlier miss, as he was sent through - legally- by Gomes, with Calvert-Lewin to his side. His attempt to take the ball round Guaita failed, as Everton wasted yet another chance to take the lead.
Late first half chances from Rodriguez and Richarlison were unable to give Ancelotti's men that all important deadlock breaker before halftime.
The Toffees started the second half with an improved enthusiasm and intensity as they looked to kick-start their European challenge.
Gylfi Sigurdsson had the first attempt as Lucas Digne's cross fell to the substitute, but his goal-bound effort was deflected behind.
It didn't take long for Everton to get the goal they longed after for the first hour of the contest. Rodriguez broke the deadlock as he was on-hand to fire Coleman's cutback past the ever impressive Guaita at the near post.
The bulging of the Gwladys street end goal was met with shrieks of relief from the Everton staff.
Top goal scorer Calvert-Lewin should have doubled the lead less than two minutes after the opener but he, like the first half, couldn't find a way past Palace's immovable No.37, as he was denied in his second one-on-one of the evening.
Palace had some late attempts of their own, as they tried to rescue a well deserved point. Ayew and Eze had long distance efforts that caused slight trouble to Olsen, but not enough to get The Eagles back in the contest.
The points should have been wrapped up 10 minuets from time when Richarlison connected with a volley after Digne teed him up inside the box, but the strikers reaction told the story, as he was left scratching his head in disbelief that Guaita, had, again, kept another one of his efforts out with a superb finger-tip save.
It would be Guaita’s heroics that would punish Everton on the night. As the game told the recurring story of the hosts’ season as their inability to kill off matches came back to haunt them. As substitute Batshuayi fired Palace level after latching onto Jeffrey Schlupp's through ball inside the box.
There was little time for Ancelotti's men to respond inside a shocked Goodison Park, as they ended the night with a disappointing point for their efforts. Meaning they are now four points off the UEFA Champions League spaces.
Everton: Olsen, Coleman, Digne, Keane, Mina, Holgate, Davies, Gomes, Rodriguez, Richarlison, Calvert-Lewin.
Crystal Palace: Guaita, Ward, Kouyate, Cahill, Van Aanholt, Milivojevic, Riedewald, Eze, Zaha, Ayew, Benteke.