The 38-year-old Brazilian hurt his left knee when making a challenge on Harry Kane and was unable to continue following lengthy treatment.
Silva, who was replaced by £70million signing Wesley Fofana, has arguably been Chelsea’s best player in recent weeks and losing the experienced defender represents a significant blow.
He went down in a heap after stopping Kane inside the Chelsea box. Silva tried to play on after being patched up on the pitch but looked in considerable discomfort and Chelsea were forced into an early change.
Chelsea are next in action against Leeds next Saturday and the Blues will hope Silva can return in time for the second leg of their Champions League last-16 tie against Borussia Dortmund on March 7. They are 1-0 down from the first leg in Germany.