Yerry Mina scored a 99th-minute equaliser as Everton picked up a potentially valuable point away to Wolves, ending their four-match winning run at Molineux.
The defender struck with seconds left to grab a 1-1 draw to give the Toffees hope after Hwang Hee-chan had given Wolves a first-half lead.
Their 69-year stay in the top flight remains in doubt and Sean Dyche’s side could still find themselves in the drop zone before next week’s finale.
Leeds go to West Ham on Sunday before Leicester’s trip to Newcastle on Monday and victory for both would lift them above Everton.
As it is, they sit two points above the Premier League’s bottom three ahead of the visit of Bournemouth next Sunday.
Dyche had told his players to ignore the noise but they struggled for long spells and again lost Dominic Calvert-Lewin to injury.
It will spawn a nervous Goodison Park, which saw its last relegation from the top flight in 1951.
Wolves’ own season of struggle – they were bottom at Christmas – will ultimately end in mid-table comfort, mainly thanks to what stands as seven home wins from 10 games since the turn of the year.
Julen Lopetegui had stressed the importance of the Premier League’s integrity, insisting his team would not roll over with matters at the bottom to be settled.
It was, though, understandable that Everton made the better start as Calvert-Lewin brushed the side-netting before nodding Alex Iwobi’s cross over.
The Toffees needed that urgency and, against a Wolves side containing six changes, they were the aggressors, although they lost Nathan Patterson to injury after 29 minutes.
Mina headed over after Daniel Bentley missed a corner and Everton were on top only to be caught on the break for the opener after 34 minutes.
The visitors were pressing on the edge of Wolves’ box but Abdoulaye Doucoure’s loose pass fell to Adama Traore who turned on the afterburners.
The forward launched into an unstoppable 70-yard dash, brushing off Amadou Onana, and when his shot was saved by Jordan Pickford, Hwang stroked in the rebound.
It was harsh on Everton but it got worse in first-half stoppage time when Calvert-Lewin, who came off in last week’s defeat to Manchester City with a groin problem, limped off.
Demarai Gray replaced him but there was no focal point and any second-half response was limited. Gray highlighted their lack of presence up front when his wicked cross through the six-yard box was missed by everyone.
Wolves had slowly gained control and Pablo Sarabia curled wide before Daniel Podence volleyed off target.
A wayward Iwobi strike was all Everton could muster until Bentley pushed Gray’s drive behind with 21 minutes left but, as time began to run out, there were few signs of a recovery.
Iwobi’s shot deflected wide and the Toffees needed Pickford to deny Matheus Nunes late on.
But they snatched an unlikely point in the ninth minute of stoppage time when Mina bundled in from close range following Michael Keane’s knockback.