His recovery period is expected to be six to eight weeks.
In 15 games with the Habs this season, the 25-year-old Finn had amassed three goals and seven points.
Armia was a first-round pick of the Buffalo Sabres - 16th overall - back in 2011.
He was acquired by the Canadiens in a trade with the Winnipeg Jets on June 30, and later signed a one-year contract.