The Spain international sustained the knock during Leeds' FA Cup fourth round win at Accrington Stanley on Sunday (AEDT).
Rodrigo is likely to miss around eight Premier League games during his absence, including the upcoming double header against Manchester United.
The 31-year-old is the club's top scorer this season, having hit 12 goals in 22 games, and a statement from Leeds on Thursday confirmed Rodrigo: "suffered bone and ligament damage as a result of a challenge and has undergone minor surgery".
The club also said that the "medical team at Thorp Arch are confident that Rodrigo will return to play by the end of March," meaning he could be back in time for April's trip to league leader Arsenal.
Jesse Marsch's men sit 15th in the Premier League, having only won two of their last 16 league games. They have not secured three points since their 4-3 win against Bournemouth on November 5.