"He brings experience and most importantly depth to our group of defensemen," Montreal Canadiens general manager Marc Bergevin said in a statement.
Drewiske scored two goals and five assists in 29 games spent withe Kings and Habs, with a plus-3 rating.
The six-foot-two, 205-pound defenceman was acquired for a fifth round pick in the upcoming NHL Draft.
The Hudson, Wisc., native has scored five goals and 20 assists in 135 NHL career regular season games.
Financial terms were not released by the club, but the deal is reportedly worth a total of $1.275 million US.