The Canadiens were seeking a veteran to play on their third defense pair.
Drewiske had one goal and three assists in 20 games and was plus-3 for Los Angeles, but had been a healthy scratch of late. He averaged 14:27 of ice time per game.
He was signed as a free agent by Los Angeles in 2008 and has four goals and 18 assists in 126 career games.
Drewiske is earning a prorated $650,000 and can become an unrestricted free agent in the summer.