Vunipola, who missed the November Test series with a calf injury, went off early in the second-half of the match and watched the rest of the game from the bench with ice strapped to his left ankle.
England's Rugby Football Union said the 28-year-old would be out of action for 10 weeks.
Vunipola was named man of the match in England's opening Six Nations victory over last year's winners Ireland on 2 February.
Eddie Jones, England's head coach, said losing Vunipola was a blow. "It is obviously very disappointing to lose Mako at this stage as he is an important player for us," said Jones. "But it does provide an opportunity for the team to grow."
Pacesetters England face unbeaten Wales in the third game of the tournament on 23 February in Cardiff.