Anthony, who ranks second on the team with 3.5 sacks, suffered the injury during Saturday's win over Central Michigan.
In the first quarter, Anthony recovered a fumble and returned it for a touchdown.
A sixth-year senior, Anthony was a key piece of the LSU defense with four tackles for loss through the first three games.
LSU has been without defensive lineman Ali Gaye, who missed Saturday's game with a minor injury.
LSU will open SEC play at Mississippi State on Saturday.