Kaprizov converted a one-timer from the right circle during a 4-on-3 advantage after Cole Caufield was called for slashing at 19:04 of the third period for Montreal.
The winner came after Nick Suzuki tied it 1-1 with 1:38 remaining in the third period for the Canadiens, knocking the puck inside the left post.
Bobby Brink scored early in the third period, and Jesper Wallstedt made 27 saves for the Wild (13-3-4), which is 7-0-2 during its point streak.
Jacob Fowler made 22 saves for Montreal (11-3-4), which extended its point streak to 11 games (8-0-3).
Fowler, the NHL Rookie of the Month in October, was 8-0-1 in his previous nine games following a 5-2 loss against the Nashville Predators in his season debut Oct. 8.
Brink gave Minnesota a 1-0 lead at 2:53 of the third when he scored unassisted on a breakaway. He stole the puck from Josh Anderson after the Montreal forward was unable to control a flip into the neutral zone, and drove in alone from his own blue line before putting a wrist shot past Fowler’s glove.
Alex Newhook appeared to score on the power play at 7:33 of the first period, but the play was overturned following a video review initiated by the NHL Situation Room. It was determined that the puck never crossed the goal line when Newhook’s shot struck the crossbar and post.
The Canadiens celebrated former goalie Carey Price’s induction into the Hockey Hall of Fame during a pregame ceremony that saw Shea Weber add Price’s name to the team’s Ring of Honor.
