The Hammers closed the gap to Chelsea in the Premier League top-four race following a superb comeback win.
David Moyes is hopeful West Ham can still nick a top-four spot in the Premier League to secure Champions League football next season.
The London side moved three points behind fourth-placed Chelsea following a 2-1 win over Burnley.
Despite Chris Wood’s 19th-minute penalty handing Burnley the lead, Michail Antonio scored twice before the half-hour mark to seal the visitors a comeback victory at Turf Moor.
But Moyes, who previously guided Everton to a top-four finish in 2004/05, accepts West Ham must win each of their last four games to stand a chance of qualifying for the Champions League.
“I’m hoping that with four games to go people still talk about us being in the Champions League,” the Hammers boss told BBC Sport.
“We want to keep going, we want to go right to the last day and see if we can sneak in. It will probably take us winning all of our games.
“I think if we made European football this year, it would be a brilliant achievement. I might admit I think I’d be disappointed now if we didn’t get there, but I don’t think it would change how we feel about the season and how we’re doing.”
West Ham return to action on Sunday in a home match against Everton.
Trips to Brighton (May 15) and West Brom (May 19) follow before Moyes’ men welcome Southampton to the London Stadium (May 23) for their final Premier League game of the season.