The Dutchman sustained the injury after scoring a hat-trick against Olympiakos in the Champions League on Wednesday and news of the blow fell hours after United were paired with Bayern Munich in the quarter-finals.
Van Persie, 30, stands to miss both legs of the tie against the defending champions and will only be able to feature in two or three of United's final Premier League games if he is sidelined for the full six weeks.
"Following further investigation, Robin van Persie has a sprained knee, which will keep him out for around four to six weeks," United said in a statement on their official Twitter account.
The news arrives at an unwelcome juncture for United manager David Moyes, who had expressed hope that the 3-0 victory over Olympiakos would prove a turning point in a wretched season.
United are currently 18 points behind leaders Chelsea in the Premier League, 12 points below the top four, and have been eliminated from both domestic cup competitions.
Speaking before the full extent of Van Persie's injury had been revealed, Moyes had confirmed that the former Arsenal striker would miss at least Saturday's trip to West Ham United in the league.
"We're just getting him re-scanned again today," Moyes said during his weekly press conference.
"We're not conclusive in everything we've seen yet, so we want to check that out and be sure. He won't be available for the West Ham game, but we'll try to give some update as soon as we can on the outcome of it."
Van Persie was stretchered off after taking a blow to the back of his knee during the mid-week game at Old Trafford, but Moyes initially played down the extent of his injury, saying it was "hopefully not too bad".
Van Persie was United's top scorer last season, scoring 30 goals in all competitions as the club romped to the Premier League title in his first campaign since arriving from Arsenal.
Injuries have restricted him to only 18 league appearances this season, but he has still contributed 11 goals.
Following reports that he had fallen out with Moyes, Van Persie insisted that he was his happy at Old Trafford in an interview with the club's match-day programme prior to last weekend's 3-0 loss at home to Liverpool.
In his absence, Javier Hernandez and Danny Welbeck will compete for the right to partner Wayne Rooney in the United attack.