The Premier League champions announced the bad news on their official Twitter page, with the Holland international cut down just two days after firing the club into the quarter-finals of the Champions League.
Van Persie's hat-trick against Olympiakos had been viewed as a possible turning point in United's poor season, but he was later carried off on a stretcher with Friday's news meaning he may not play again until May.
He is almost certain to miss the Champions League last-eight tie with Bayern Munich, which he single-handedly set up.
A tweet from United read: "Following further investigation, Robin van Persie has a sprained knee, which will keep him out for around four to six weeks."
Manager David Moyes had earlier suggested that there could be bad news regarding the former Arsenal man.
"We're getting Robin re-scanned again today," the Scot had said.
"We're just not quite conclusive on everything we've seen yet. We want to just check that out and be sure. He won't be available for (Saturday's) game (against West Ham). But we'll try to give you some update when we can on the outcome of it.
"We've got a lot of games coming up and we need all our players fit and available."
As well as the West Ham and Munich games, Van Persie, who has scored 17 goals this season, will sit out the derby against Manchester City on Tuesday.
It is the sort of news that will disappoint Moyes, who on Friday spoke of his shame at the some of the results he has overseen this season.
Moyes enjoyed what he admitted was his best 90 minutes as United manager on Wednesday when his team beat Olympiakos.
His reward - if it can be termed as such - is a last-eight clash against holders Bayern Munich.
Before he starts plotting how to beat Pep Guardiola's team of superstars, Moyes must concentrate on a crucial four days ahead.
United visit West Ham before taking on Manchester City at Old Trafford on Tuesday. If United drop a point in either game, they will be guaranteed to end the season with their worst points total in Premier League history.
However, at least Moyes can go into the double-header with his head held high after Wednesday's win.
"I think the big thing for me was that I could look at the supporters in the face after the game on Wednesday night because I thought we had given them something to shout about," Moyes said.
"We had nothing to be ashamed of with our performance. In some other games I have felt ashamed because the team haven't played well enough and I've not got a team out which has got a good result."
The booming atmosphere inside Old Trafford on Wednesday was reminiscent of the many glorious nights enjoyed by Moyes' predecessor during his 26-year reign.
But one thing which Sir Alex Ferguson achieved which has so far eluded Moyes has been consistency.
Time after time Moyes has seen his team achieve encouraging results but then flop.
United went into the new year on the back of six consecutive wins, but they followed that up with three straight defeats.
It was a similar story in November when Moyes' men won three on the bounce before going three weeks without a league win.
So Moyes was cautious when given the opportunity to describe the win over Olympiakos as a turning point in what has been a severely disappointing season.
"I have been asked a similar question many times this season and I have always said we will build on it but on several occasions we have not been able to do that," Moyes added.
"We need to make sure that we get a bit more consistency.
"It's going to take time, we're not going to be able to do everything immediately but my job over the coming years is to make sure that we get rid of (inconsistency)."
Antonio Valencia was another walking wounded member of United's victorious XI.
The winger suffered a huge black eye following a clash of heads with Joel Campbell, and the Ecuadorian is therefore doubtful for Saturday's game.
"Antonio has a real bad eye which is closed over," Moyes said. "It is not quite settled down yet so we will need to keep an eye on it, if you'll pardon the pun!"