Alzner, 30, has played eight games for the Canadiens this season, registering one assist and a plus-2 differential.
A native of Burnaby, BC, Alzner signed with the Canadiens as a free agent on July 1, 2017. He suited up for all 82 games last season, collecting one goal for 12 points.
The 11-year NHL veteran boasts 681 games of NHL experience, amassing 20 goals and 130 points along the way.
Alzner was the fifth-overall pick of the Washington Capitals in the 2007 NHL Draft.