West Ham have continued to remain in contention for a top four finish despite being expected by many to fall away and further added fuel to their chances by edging out Burnley 2-1 on Monday night.
The Hammers fell behind through a Chris Wood penalty, but hit back through a quick Michail Antonio brace, with his goals coming in the 21st and 29th minutes.
The three points at Turf Moor moves West Ham up to fifth in the standings and just three points behind fourth placed Chelsea with four games to play.
Moyes believes that his men will need to win their remaining games to be able to finish in the top four, but is hopeful that regardless they will have done enough to secure at least a Europa League place.
“We didn’t deserve to go 1-0 down. I thought we started well and were the better team. We gave away a cheap penalty kick”, he told BBC Sport post match.
“Great resilience, credit to the players that we got the goal back quickly.
“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.
“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.”
West Ham’s remaining games in the Premier League this season are Everton (home), Brighton (away), West Brom (away) and Southampton (home).