Vitalii Mykolenko crossed from the left, James Tarkowski headed the ball down, Kiernan Dewsbury-Hall lashed it into the back of the net and for a moment it felt as if West Ham were sliding towards the Championship.

It was 1-1 at the London Stadium, Tomas Soucek’s opener on his 250th appearance for the east Londoners cancelled out. But added time is proving crucial in this relegation battle. Last week it was Georginio Rutter shattering Tottenham with a late equaliser for Brighton. This time it was Callum Wilson popping up in the 92nd minute to score his ninth goal in his last eight games against Everton and maintain West Ham’s two-point advantage over Spurs, who remain in 18th place despite their win over Wolves.

Wilson, who had come on with West Ham trying to hold out in the face of stiff Everton pressure, had chipped in again. It was the veteran striker who kickstarted West Ham’s revival when he scored a last-minute winner against Spurs in January. He was at it again here, rescuing Nuno Espírito Santo’s side just when it seemed they were heading back into the relegation zone. David Moyes, who marked the 750th Premier League game of his career with a defeat that derails Everton’s hopes of European qualification, was left to rue poor finishing and iffy defending against his old side.

With the margin for safety slipping away after Nottingham Forest’s destruction of Sunderland, the tension threatened to mess with West Ham’s focus. It was easy to imagine Hammers fans thinking nervously of events at Molineux. Would Wolves do them a favour? Or would the balance of power swing Tottenham’s way? And could anyone think of a more unbearable way to spend a Saturday afternoon?

The anxiety contributed to a disappointing first half, with the atmosphere as flat as the football. There was silence as Everton took control during the first 10 minutes. West Ham had no flow. The service to Pablo Felipe and Taty Castellanos was poor and Jarrod Bowen was tentative with one early chance.

Not that Everton’s attack was much better. Moyes, back in east London for the first time since his departure two years ago, saw little to excite him on his 63rd birthday.

West Ham needed to assert themselves at the start of the second half. They looked to the left, to Crysencio Summerville, who was the liveliest attacker on show. The Dutchman’s pace worried Everton, with James Tarkowski, James Garner and Jake O’Brien both booked for chopping him down.

Summerville dusted himself down and came back for more. He created West Ham’s first proper opening, his cross dropping to Castellanos, whose shot was saved by Pickford and turned behind for a corner.

Now, having delivered a series of disappointing set pieces during the opening period, West Ham finally got one right. Bowen produced an inswinging cross from the right and Everton, who conceded from a late corner when they lost last week’s Merseyside derby, were punished again by Soucek taking advantage of soft defending at the near post by rising to glance a header past Pickford.

There was still a long way to go. Moyes reacted by introducing replacing Dwight McNeil with Tyrique George, who made a difference on the left. West Ham sat back as they closed on a third consecutive clean sheet, inviting pressure. Soucek hacked clear after a mistake from El Hadji Diouf; then he headed Barry’s header on to the bar after George’s cross found the Frenchman unmarked in the six-yard box.

Everton continued to press. They screamed for a penalty when Mateus Fernandes used a hand to scoop the ball away from Barry. Somehow there was no intervention from Stockley Park but their luck soon ran out. Mykolenko crossed, Tarkowski nodded the ball down and Dewsbury-Hall lashed in the equaliser.

With Spurs 1-0 up, West Ham were heading into the bottom three. Eight added minutes offered hope, though. There were 92 minutes on the clock when Diouf’s deep cross found Everton all at sea. Bowen headed the ball across and Wilson finished Everton off. This is going to the wire.