The Netherlands international missed four matches with a groin problem before making his return to action in the 1-0 loss to Newcastle United, playing the full 90 minutes.
Van Persie also featured on Tuesday night against Shakhtar Donetsk, coming off the bench to assist Phil Jones' winner, and it was in the process of taking that corner that he sustained the injury.
"Robin has picked up a thigh strain and will be out for a month," Moyes told reporters. "He took the corner we scored the goal from against Shakhtar and felt his thigh.
"So we're going to lose Robin, it looks like, for a month, with a thigh strain."
The injury means last season's Premier League top scorer will miss the busy Christmas period, though United does not face a top-10 side until Tottenham visits Old Trafford on Jan. 1.
Moyes will have to make do without the Dutch forward for Sunday's trip to Aston Villa, where he hopes to get back to winning ways and end the club's four-game winless run.
"We want to play well," the Scot added. "We've not done as well in the league as we should have. We want to put it right.
"We've got a lot of games coming up and can pick up points. But we need to do it, not talk about it."