Early in what proved to be a painful Merseyside derby for Liverpool, Van Dijk was the subject of a reckless lunge by Jordan Pickford that saw the Everton goalkeeper clatter into the extended right leg of the Netherlands defender.
Van Dijk roared in agony and, tellingly, stayed down.
There was no official word from Liverpool on the potential length of his absence, but Van Dijk reportedly left hospital on crutches in the evening.
Van Dijk might have been marking Keane and Calvert-Lewin for both of those Everton chances.
For Southamptonās second goal, Kurt Zouma underhit a back pass before goalkeeper Kepa Arrizabalaga made two errors in a matter of seconds to allow Che Adams to score.