After the two teams combined to score four times during the first three rounds of the tiebreaker, Jhonas Enroth and Price shut the door for the next five rounds until Hecht's forehand-to-backhand flip finally won it for Buffalo.
The successful shootout capped a Buffalo comeback that saw the Sabres rally from a 2-0 deficit, thanks to a second-period tally by Drew Stafford and Jason Pominville's equalizer 6 minutes into the final frame.
Enroth stopped 32-of-34 shots for the Sabres.
Max Pacioretty and Benoit Pouliot scored for the Habs, who blew a chance to move into first place in the Northeast Division after Boston's loss to Toronto Tuesday.
Price finished with 30 saves for Montreal.