The 24-year-old forward suffered the injury on Saturday when he landed awkwardly during the Saints' 4-1 defeat at Manchester City in the Premier League.
Rodriguez will now miss the rest of the season as well as this summer's World Cup in Brazil, where he was expected to be part of the England squad.
"Jay has asked us to pass on his gratitude to everybody who has shown him their concern and sympathy," executive director Les Reed told Southampton's official website.
"Jay is naturally disappointed, but is determined to get back playing for Southampton as soon as possible."
Rodriguez had scored 17 goals in 39 games for Southampton this season before being injured.