The Hammers beat Burnley 2-1 to maintain their pursuit for a top-four place in the Premier League and Moyes thinks his team will need to win all of their last four matches to qualify for Europe's top competition.
He told BBC Sport: "I'm hoping that with four games to go people still talk about us being in the Champions League.
"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."
Moyes also confessed that he would feel a sense of disappointment if his team doesn't qualify for at least the Europa League after defying the odds to stand on the brink of a top-six finish.
He said: "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."
Moyes was thrilled by his team's performance as they moved up to fifth place and the coach is determined to enjoy the experience.
He said: "The team played really well. We won 2-1 but I felt it was a lot more than that. We rode our luck at the end but we should have been out of sight.
"I’d love to be in Europe. It would be an incredible achievement. We’re not far away from the Champions League positions. We’ve got to keep believing.
"Four games to go, it’s great. I want to smile about it. I’ve been here before – we’ll probably have to win the four games. If we played the way we played tonight we’ll give most teams a game."