Jake O’Brien’s 91st-minute equaliser saw David Moyes extend his, and Everton’s, unbeaten run to nine games with a 1-1 draw against his former club.
Tomas Soucek had looked likely to settle a largely forgettable contest and clinch a third victory in four games for a West Ham United side that are slowly starting to show signs of life under new manager Graham Potter.
The Czech international struck on 67 minutes as a game that had been low on quality finally burst into some degree of life.
Jarrod Bowen created the goal, spotting his team-mate in space in the Everton area and threading a perfect pass, and Soucek did the rest with an impressive finish beyond the dive of Jordan Pickford and into the far corner.
Pickford was soon required to make further saves from a Bowen shot and Soucek header as both sides suddenly discovered some urgency.
But those misses looked costly in added time when Carlos Alcaraz swept the ball wide to Idrissa Gueye whose chip into the six-yard area was firmly headed in by O’Brien.
Before that, a first half low on chances, and attacking intent, ended on a note of controversy, with Everton first awarded, then denied, a penalty by referee Darren Bond.
The original 44th-minute decision followed Beto going to ground, apparently tripped by Konstantinos Mavropanos, before VAR intervened and Bond, correctly, changed his mind after looking at his video screen.
That decision understandably annoyed home supporters but, if not for Pickford, they could have found themselves down at the interval.
On 12 minutes, Everton defender Jarrad Branthwaite connected with a James Ward-Prowse corner, heading goalwards and forcing his goalkeeper into a flying reflex save.
And just before the half-hour, Bowen skilfully teed himself up, juggling the ball until it sat up for an 18-yard volley which Pickford turned away with a one-handed stop.
