After beating Burnley on Monday the Hammers boss declared that his side would have to win all four of their remaining games to sneak into the Premier League's top four.
Dominic Calvert-Lewin burst that bubble with the only goal in a tight game at the London Stadium, with the hosts dominating the ball but failing to muster a shot on target.
Said Benrahma and Jarrod Bowen both missed clear cut chances to get West Ham back in the game, but could not convert leaving the Hammers five points behind fourth-place Leicester with three games to play.
"Our chances of the Champions League are looking slim but we'll keep fighting," said Moyes.
"I'd set my target and I knew where we had to go, and defeat might be costly for us but you never know in football, we might be saying something different in time.
"We have to keep winning and try and do that, I don't think we deserved to lose today so we have to try and pick it up and get the next win under our belt as soon as we can.
"If we qualify for Europe it will be an incredible achievement no matter what tournament we're in."