Spurs looked set to escape the bottom three with just minutes remaining after they took the lead against Wolves and West Ham conceded an equaliser against Everton.
But substitute Callum Wilson scored a winner in the 92nd minute at the London Stadium to restore West Ham’s two-point cushion, with just four games to go.
Roberto De Zerbi’s Tottenham travelled to already-relegated Wolves without a Premier League win since December but were blunt in attack.
But Joao Palhinha’s late goal secured a 1-0 win and travelling Spurs fans celebrated wildly when they learned that Everton had levelled in the 88th minute.
At that stage Spurs were out of the bottom three.
But West Ham were not finished, with Wilson adding a remarkable postscript.
Wolves and Burnley are already down but the race to avoid the final relegation spot has effectively become a two-horse race between Tottenham and West Ham.
Nottingham Forest hammered Sunderland 5-0 on Friday to pull well clear of the drop zone.
Liverpool beat Crystal Palace 3-1, getting the better of their opponents for the first time in four meetings this season to climb into fourth place above Aston Villa, who earlier lost 1-0 at Fulham.
Liverpool lost the season-opening Community Shield against Palace on penalties before defeats in the Premier League and League Cup.
Alexander Isak, who has recently returned to action after fracturing his fibula, scored his first goal for Liverpool since December.
Andy Robertson doubled Liverpool’s lead after a slick counter-attacking move.
Daniel Munoz halved the deficit in the second half but Florian Wirtz scored a last-gasp third to restore the home side’s two-goal cushion.
Arne Slot’s men are now near-certainties to qualify for next season’s Champions League, which would be some consolation after a miserable title defence.
They are level on 58 points with Villa and Manchester United — with all three teams eight points clear of Brighton.
In the early kick-off on Saturday, Fulham beat Aston Villa to keep their hopes of qualification for Europe alive.
Ryan Sessegnon scored the only goal of the game shortly before half-time after Morgan Rogers and Ollie Watkins had gone close for Villa.
Fulham are now 10th, level on 48 points with neighbours Chelsea and just two behind sixth-placed Brighton in the scrap for places in the Champions League, Europa League and UEFA Conference League.
Arsenal aim to retake top spot from Manchester City when they host Newcastle in the evening kick-off.
The Gunners have seen their nine-point lead vanish in the past two weeks and they are now behind Pep Guardiola’s men on goals scored, with both sides level on goal difference.
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