Price turned aside all 14 shots he faced in the third period and another in overtime before the game moved to a shootout.
Lars Eller bested Tampa Bay's Ben Bishop in the first round of the tiebreaker, and Price then secured the victory with scintillating saves on Nikita Kucherov and Teddy Purcell.
Tomas Plekanec scored in regulation as the Canadiens took a third win in four games.
Martin St. Louis got the only goal for the Lightning, who had their five-game winning streak snapped.
Bishop finished with 20 stops.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys