Boss David Moyes has urged West Ham to finish the job and clinch a Europa League spot after a 3-1 win at West Brom.
The Hammers need a point in Sunday’s final game of the season at home to Southampton to qualify for the Europa League.
Angelo Ogbonna and Michail Antonio netted in the final eight minutes to complete a comeback victory at The Hawthorns while, after defeat, the Baggies confirmed boss Sam Allardyce will leave at the end of the season.
Tomas Soucek’s own goal had given the Baggies the lead during a nightmare start for the Hammers after Declan Rice missed a third-minute penalty.
Soucek levelled before the break, though, and the Hammers’ late show left them sixth in the Premier League, three points ahead of Tottenham and Everton.
Moyes said: “It is (close) and I’m delighted but it’s not done. I’ll go away, we’ll sit back and enjoy it but we know we have to be in charge and win the game, if we win the game there’s no danger.
“We have been playing really well, tonight we didn’t but we found another way of winning. We’re beginning to get a group who can find ways of winning depending on what type of game it is.
“It showed a lot of big heart from the players to keep at it. We have equalled a club record for points in the Premier Legaue, one more point would beat that and get us into Europe as well.
“It was a different type of win, it was a really tough game for us. The way West Brom played was difficult to play against. It was a good victory for us in the end.”
The Hammers had already lost Lukasz Fabianski to a knee injury in the warm-up and their start got worse when Rice hit the post from the spot after three minutes when Antonio was brought down by Sam Johnstone.
They then found themselves behind after 27 minutes when Soucek glanced Matheus Pereira’s corner into his own net.
The midfielder atoned for his error in first-half stoppage time, tapping in Said Benrahma’s cross, to give the visitors a platform for their second-half recovery.
Darren Randolph needed to save well from Semi Ajayi but, after Aaron Cresswell’s free-kick hit the post, West Ham won it with two goals in the final eight minutes.
Ogbonna made it 2-1 when he headed in Cresswell’s corner and Antonio grabbed his 10th goal of the season after the Hammers hit Albion on the break.
Baggies boss Allardyce, who will step down after Sunday’s trip to Leeds, told Sky Sports: “Michail Antonio’s statement which I just heard, he said we had big men who just play in the channel. He’s just insulted me and my team. I know he’s a nice lad but it was disgusting what he said.
“Our front two are five foot 10 and five foot nine, I don’t know what big men he saw running in the channel.
“If you thought that as long ball then someone is deluded.”
The Hawthorns welcomed 5,371 fans for the first time in over a year and Allardyce believes the supporters helped his team’s performance.
He added: “I know the atmosphere as well as anyone else at West Brom over many years of visiting as a manager.
“Even though we got beaten, the fans made a huge difference to the players’ performance. They got a goal from a corner and it was a shame we ultimately lost to that goal.”