Michael Bournival added an empty-net goal for Montreal (4-1-0) with 59 seconds left.
Andrew Shaw ended Price's shutout bid with 5:53 left on a breakaway on Chicago's 29th shot.
Price was sharp for a second straight complete game as he recovers from a knee injury sustained in the Eastern Conference finals. The Canadiens lost 2-0 to Washington 2-0 on Sunday, but Price didn't allow a goal until 1:29 was left.
Antti Raanta, expected to back up Corey Crawford again this season, made 16 saves for Chicago (2-3-0).