Dominic Calvert-Lewin's instinctive finish ensured a depleted Everton earned a 1-1 draw against West Ham at the London Stadium.
Issa Diop headed the hosts in front five minutes before the break on his return to the side, but Calvert-Lewin responded immediately with his 11th goal of the season to restore parity (44) from close range.
Robert Snodgrass came close to regaining the lead for David Moyes' side as his deflected shot forced a fine save from Jordan Pickford midway through the second period.
West Ham: Randolph (6), Zabaleta (6), Diop (7), Ogbonna (6), Cresswell (6), Snodgrass (7), Noble (7), Rice (6), Fornals (6), Lanzini (6), Haller (6).
Subs: Masuaku (6), Ajeti (n/a).
Everton: Pickford (7), Coleman (7), Mina (7), Holgate (8), Digne (7), Walcott (6), Davies (5), Delph (6), Bernard (5), Calvert-Lewin (7), Kean (6).
Subs: Sidibe (7), Niasse (5), Gordon (6).
Man of the match: Mason Holgate.
How the points were shared after the protest
Hundreds of West Ham fans staged a protest against the club's current owners ahead of the game, with Saturday marking 10 years in charge for David Gold and David Sullivan - but those who were still outside during the opening 15 minutes had not missed much.
Everton manager Carlo Ancelotti had failed to win any of his four previous Premier League meetings with Moyes in his previous stint in the competition, and without their talisman Richarlison, the onus was on the hosts to take the game to the under-strength visitors.
Moyes admitted this week he had talked to Everton about a potential return to the club he managed for 11 years before they appointed Ancelotti in December, and his side created the better opening chances. Snodgrass had already drawn a routine save from Pickford when Mark Noble's inviting cross was glanced a yard wide by Sebastien Haller.
Moments later, Snodgrass was correctly ruled offside as he fired beyond Pickford, but West Ham really ought to have broken the deadlock when Noble slipped in Haller down the right only for the Frenchman to see his shot kept out by the feet of Pickford.
Darren Randolph was straight back into the starting line-up in place of the injured Lukasz Fabianski following his return to the club. Fabian Balbuena was replaced by Issa Diop while Pablo Fornals and Robert Snodgrass started in place of Arthur Masuaku and the absent Felipe Anderson.
Everton made four changes to the side that beat Brighton as Seamus Coleman, Yerry Mina, Fabian Delph and Moise Kean came in for Djibril Sidibe, Michael Keane, Gylfi Sigurdsson and Richarlison.
Everton, in contrast, were struggling to create opportunities in the absence of Gylfi Sigurdsson. It took until the 32nd minute for the Toffees to test Darren Randolph as Theo Walcott scuffed his shot from Lucas Digne's cross into the gloves of the returning Hammers' goalkeeper.
The game sparked into life five minutes before the break as West Ham played on Everton's vulnerabilities at defending set pieces. Ancelotti has brought back a form of zonal marking, but it proved ineffective as Diop rose to glance home Snodgrass' delivery - the ninth goal the visitors have conceded from a deal ball this season.
Cajoled into showing greater urgency, Everton immediately equalised from the same means, as Digne's corner was flicked on by Mason Holgate and headed in from close range by Calvert-Lewin. But West Ham could so easily have led at the break as Pickford brilliantly kept out Pablo Fornals' header from Pablo Zabaleta's cross.
Ancelotti made a change at the interval as Anthony Gordon was introduced for his first Premier League appearance, and the 18-year-old very nearly made an instant impact as he squandered a glorious chance, blazing over after Randolph had palmed Theo Walcott's cross into his path.
Diop then had to be alert to produce a vital last-ditch challenge on Calvert-Lewin at the near post before Tom Davies headed wide from Digne's ensuing corner.
But West Ham regained a modicum of control and were unfortunate not to regain the advantage when Pickford very nearly carried the ball over the line from Haller's looping header, denying the £45m summer signing only a third goal since his arrival.
England's No 1 atoned for that lapse in concentration to brilliantly deny Snodgrass from his deflected shot with 20 minutes remaining, but both sides appeared happy in the end to settle for a point.
There was a brief skirmish in the closing stages as Albian Ajeti clashed with Holgate, pushing his head into the face of the defender, but following a VAR check for a potential red card for violent conduct, no further action was taken against the West Ham substitute.
What the managers said…
West Ham boss David Moyes: "We didn't take our chances in the first-half but Jordan made a couple of good saves. Everton are in good form over the last six games, so I knew it was going to be tight. With both sides missing players, I knew it was going to be nip and tuck.
"Every point is really important at the moment. We're in a tight part of the division at the moment but a draw just keeps things going.
"I want the supporters to stay right behind the players. They've been really good since I've been in. There's been small improvements that we saw today and in certain areas, but now we need to give them a team to give them something to shout about."
Everton boss Carlo Ancelotti: "It's a good point - of course, we didn't deserve much from the first half. We did a lot of mistakes and West Ham played better than us. In the second half, we played differently but it's a good point. In the end, we deserved the point.
"I think we can and must improve this season. Some part of the game is good but some parts aren't and we have to show more intensity in games. We have to be more precise when we have possession. We don't have to be too shy when we play the ball."
Man of the match - Mason Holgate
Everton arrived in the capital nursing several injuries, with both Holgate and Yerry Mina carrying knocks, but the young Englishman produced a mature display to underline the defensive improvements under Ancelotti.
The 23-year-old began the season as the club's third-choice centre-back, but with each passing game, it now appears that Mina and Michael Keane are competing to be his partner.
Holgate made more clearances (seven) and more blocks (two) than any other player while his four tackles were bettered only by West Ham's Mark Noble. For good measure, it was his flick on that lead to Calvert-Lewin's equaliser.
'The game lacked killer quality'
Analysis from Sky Sports' pundit Charlie Nicholas:
"It lacked quality in the last third of the pitch for both sides. West Ham should have been two up in the first half but went to sleep and got lazy on it. The second half just lacked the killer cross and Pickford made a couple of good saves.
"I feel West Ham should've won it, but to be fair to Ancelotti, he does not hesitate in making changes. He took off Bernard at half-time and brought on Gordon, who did OK.
"He made three quick substitutions and that was a sign of him saying, 'this is nowhere near good enough, I want better'."
- West Ham have won just one of their last 12 home Premier League games against Everton (D4 L7), a 3-1 victory in May 2018.
- Everton have conceded in 11 consecutive away Premier League games for the first time since October 2010-March 2011 under David Moyes (also a run of 11).
- West Ham have dropped more points from winning positions (17) than any other Premier League side in 2019-20.
- Everton remain unbeaten in their last eight Premier League matches against London sides (W2 D6), their longest run without defeat against such opposition in the competition since November 2008-May 2009 (a run of 9).
West Ham face Leicester at the King Power Stadium on Wednesday; kick-off 7.30pm. The Hammers are in FA Cup fourth round action next weekend at home to West Brom.
Everton host Newcastle at Goodison Park on Tuesday - also at 7.30pm - before they travel to face Watford on February 1.