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.
Hedman cut the deficit to 3-2 on an in-close goal that stood up following a video review with 1:43 remaining. Salo tied it from just inside the blue line with 43.3 seconds left.
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.
Tampa Bay, which opened the season 6-1, has since lost five in a row since then (0-4-1).
After Subban scored on a blue line shot at 8:21 of the second, Moen made it 3-0 from the low slot with 1:06 to go in the period. Moen snapped his 21-game goal drought, dating to Dec. 31, 2011, against Florida.
Gionta put Montreal ahead 1-0 when he tipped a pass through the crease from Tomas Plekanec past Lindback during a 5-on-3 power play with 43 seconds left in the first.
The Canadiens had a couple of quality scoring chances early in the first.
Lindback also made a lunging blocker stop on Markov's rebound attempt.
Price stopped an in-close redirection by Vincent Lecavalier late in the first, and had a glove save on Steven Stamkos' shot from the right circle during a second-period power play.
Montreal's Lars Eller had two assists.
NOTES: Tampa Bay LW Ryan Malone sat out his second straight game and could miss three to four weeks because of an undisclosed lower-body injury. ... It was Gionta's first goal in nine games. ... Lightning RW Richard Panik, recalled from Syracuse of the AHL on Monday, made his NHL debut. ... Armstrong has gone 24 games without scoring a goal. He assisted on Moen's goal for first point in 17 games.