BURNLEY - A 2-1 win for West Ham against Burnley has put them within touching distance of Champions League football next season.
The Hammers rose to fifth in the Premier League table, three points behind Chelsea in the final Champions League spot with four games left to play.
ALSO READ: Everton missed big opportunity in race for Europe, says Carlo Ancelotti
West Ham still have to face Everton (8th), Brighton & Hove Albion (14th), West Bromwich Albion (19th) and Southampton (15th), while Chelsea have a tougher run-in, taking on leaders Manchester City, Arsenal (9th), Leicester City (3rd) and Aston Villa (10th).
"I'm hoping with four games to go, people still talk about us and the Champions League. We want to take it down to the final day and try to sneak in," Moyes told BBC Sport.
ALSO READ: Pep Guardiola declares love for departing Sergio Aguero after Palace special
"It will probably need us to win all four games. Europe would be an incredible achievement. I think I'll be disappointed if we don't get there now.
"Most of the games this season we've played very well. We've missed some players through injury which has affected us. We'll keep going, we're not going away."
Moyes welcomed back Michail Antonio from injury, and the striker responded by netting both goals for the Hammers.
"Michail Antonio made a great impact. He's come back in good condition, good shape ... he's a big influence in the team and we're glad to have him back," Moyes said.
With Burnley nine points above the relegation zone, their survival in the Premier League is not yet mathematically ensured but manager Sean Dyche believes they are trending in the right direction.
"They (West Ham) are chasing Champions League spot and we are trying to look after ourselves. We didn't go too far wrong. Our level of performance has been good," he told Sky Sports.