The visitors fell behind to a penalty from Chris Wood but a quick-fire double from Michail Antonio gave West Ham a priceless win.
West Ham’s victory over Sean Dyche’s men moved them up to fifth place, three points behind Chelsea, who occupy the final Champions League spot, with four games remaining.
Moyes reckons that West Ham will have to remain faultless in their remaining games if they are to qualify for the Champions League.
"I'm hoping that with four games to go people still talk about us being in the Champions League," Moyes said to 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 United’s four remaining games are against Everton, Brighton and Hove Albion, West Bromwich Albion and Southampton.
