In the second, the Canadiens got into a bit of penalty trouble, taking three minor penalties in the period. But the Habs' special teams were up to the task, killing off all three Detroit power plays.
Neither team lit the lamp in the middle frame, with Detroit outshooting Montreal 14-9.
In the third, the Red Wings tied the game just past the halfway mark of the period when Lucas Raymond one-timed it in past Allen for his fifth of the season.
That would be the last goal before regulation time ran out, meaning this game would be decided in extra time, although overtime didn't determine a winner.
In the shootout, Cole Caufield and Nick Suzuki scored for the Habs, and with only David Perron finding twine for the hosts, Montreal took the game.
The Canadiens traveled home immediately following the game. They face the Vancouver Canucks at the Bell Centre on Wednesday night; for tickets to that game,
click here
.