David Moyes’ side were denied a penalty late in the first half when Beto was fouled by Kostas Mavropanos before a VAR review overturned the decision.
They found themselves behind with barely 25 minutes remaining as Tomas Soucek was able to squeeze a shot into Jordan Pickford’s far corner.
But at the end of normal time, they salvaged a point courtesy of a well-worked move as Jake O’Brien powered home from Idrissa Gueye’s volleyed delivery.
The Blues have moved 17 points clear of the Premier League’s relegation zone ahead of the Merseyside derby after March’s international break.
Winning the Premier League’s Manager of the Month award is often a kiss of death for its latest recipient and David Moyes almost became the latest victim.
Fresh from being crowned February’s outstanding candidate on Friday, the Everton boss hoped to disprove that theory for the visit of West Ham United.
But Moyes’ former employers threatened to condemn him to the end of an impressive eight-game unbeaten stretch since returning to Goodison Park.
Until Jake O’Brien’s last-gasp equaliser, the Blues could have few complaints about that outcome after labouring for large parts of the second 45 minutes.
They started strongly yet failed to take their chances and were punished after the interval as Tomas Soucek put the Hammers ahead from one of his own.
One thing about Moyes’ Everton, though, is that they never know when they are beaten and belatedly rallied to salvage a point from the jaws of defeat.
His ability to avoid the Manager of the Month ‘curse’ has also hepled set his charges up to tackle a challenging month ahead in the best position possible.
Moyes was afforded a rare luxury in his first reunion with the Hammers.
Since he succeeded Sean Dyche in mid-January, the Scot had been forced to contend with limited squad options due to Everton’s mounting injury woes.
Seasoned Goodison match-goers have often seen the inclusion of two goalkeepers among their substitutes as symptomatic of that dearth of options.
Yet third-choice stopper Asmir Begovic found himself consigned to surplus following the long-awaited returns of Armando Broja and Youssef Chermiti.
The pair spent a combined nine months on the sidelines, with Broja laid up since early January while Chermiti had not featured at all in this campaign.
Gauging their respective readiness for the cut and thrust of action may have proved difficult with just 11 minutes of normal time to influence proceedings.
Both, however, offered promising cameos with Chermiti regularly seeking to test Alphonse Areola while Broja showed flashes in his own comeback.
It has been a long time since Everton were able to deploy transformative options from the bench but things appear to finally be looking up on that front.
Thomas Tuchel’s maiden England squad was always set to divide opinion.
His decision to recall Jordan Henderson, at 34, raised eyebrows as did some of the other inclusions for the World Cup qualifiers with Albania and Latvia.
Moyes responded to Jarrad Branthwaite’s omission for the Three Lions by challenging him to force a way back into the German’s long-term plans.
On his first outing since being downgraded to Lee Carsley’s under-21 side, the 22-year-old’s case for the defence saw him endured mixed fortunes.
A first involvement saw Jordan Pickford inadvertently forced into an early save to prevent Branthwaite’s misplaced header from a corner finding his own net.
But as the hosts grew into the game, the Carlisle native demonstrated his astute qualities by superbly marshalling of Aaron Wan-Bissaka out of play.
This display will not catapult him onto Tuchel’s radar immediately but there are signs that Branthwaite could be back in favour sooner rather than later.