Allen was chasing the Wizards' Nick Young around a baseline screen and went down in a heap. He attempted to walk it off but quickly signaled to coach Doc Rivers for a substitution before limping gingerly toward the bench.
Allen had seven points on 3-of-5 shooting before leaving with the injury.
The Celtics are already without point guard Rajon Rondo, who missed his second straight game with a right wrist injury.
Allen entered Sunday's contest averaging a team-best 15.5 points per game, along with 3.2 rebounds and 2.5 assists.