With a little over seven minutes left in the frame, though, centerman Pierre-Luc Dubois responded for the Blue Jackets and squared the scoreline by beating Carey Price on a shot from the slot.
After 40 minutes, the Canadiens held a 24-20 advantage in the shot department.
Columbus retook the lead with just over 13 minutes remaining in regulation time when forward Emil Bemstrom capitalized on a lucky bounce off the glass behind Price that ultimately deflected into the net off the seven-time All-Star.
Boone Jenner put the Blue Jackets ahead by a pair when he scored on the power play.
With 5:27 to go, Dubois scored his second goal of the game to put the Blue Jackets ahead 5-2.
That's the way things finished up.
The Canadiens will be back in action on Wednesday night against the Ottawa Senators at the Bell Centre.