Victor Hedman scored twice and Sami Salo added a power-play goal during the final six minutes of regulation as the Lightning tied it at 3.
Tampa Bay's Anders Lindback made a series of strong saves during a Montreal power play during overtime. He finished with 31 saves, including six in the extra session.
Price was bidding for his 17th NHL shutout and first since Feb. 11, 2012, when Hedman scored with 5:56 to play.
Brian Gionta, P.K. Subban and Travis Moen scored for the Canadiens, who were 0-2-1 over their previous three games. Price stopped 21 shots.
Stay Connected
NHL.TV™
NHL GameCenter LIVE™ is now NHL.TV™.
Watch out-of-market games and replays with an all new redesigned media player, mobile and connected device apps.
Introducing the new official NHL App, available for iPhone, iPad and Android smartphones and tablets. A host of new features and improved functionality are available across all platforms, including a redesigned league-wide scoreboard, expanded news coverage, searchable video highlights, individual team experiences* and more. The new NHL App on your tablet also introduces new offerings such as 60fps video, Multitasking** and Picture-in-Picture.
*Available only for smartphones
** Available only for suported iPads