The 30-year-old has an assist in eight games this season.
He had a goal and 11 points in 82 games with Montreal last season after signing a five-year, $23.125 million deal in the 2017 offseason.
The move comes as top defenseman Shea Weber nears his return from offseason knee surgery.
Alzner was the fifth overall pick of the Washington Capitals in the 2007 draft.
The 11-year veteran has 20 goals and 110 assists in 681 regular-season games.