Ryan Malone scored in regulation time for Tampa Bay (13-5-0), while Ben Bishop stopped 28 shots.
Daniel Briere was the lone goal scorer for Montreal (9-8-2).
Malone put Tampa Bay on the scoreboard at 5:00 of the first when he deflected an Eric Brewer shot from the point.
Malone, who was shoved by Canadiens defenceman Douglas Murray seconds before Brewer's shot, deflected the puck while falling to the ice. The goal was his third of the season.
Montreal couldn't take advantage of three power-play opportunities in the first period. They finished the game 0 for 3 with the man advantage.
The Habs, who were outshot 13-5 in the second frame, dodged a bullet late in the period when Radko Gudas beat Price with a shot from the point. The goal was disallowed when video replay suggested Tyler Johnson had obstructed Price on the play.
Briere got the equalizer at 15:22 of the third when a Max Pacioretty shot went off his thigh and past Bishop. Briere was back in the Canadiens' lineup for the first time since suffering a concussion on Oct. 19. The 36-year-old, who missed 10 games, played on a line with Michael Bournival and Rene Bourque.
Overtime solved nothing.
The Lightning were without their leading goal-scorer Steven Stamkos, who broke his right leg after crashing into the net against the Bruins on Monday. Stamkos underwent surgery Tuesday afternoon and is expected to miss several months of play.
Rookie J.T. Brown, who was recalled from Syracuse (AHL) after Stamkos' injury, recorded one shot in 14:37 of ice time. Brown played five NHL games with the Lightning in 2012.
Notes: The Canadiens have only won once in their last six games. They beat the New York Islanders 4-2 on Sunday. … David Desharnais and George Parros were both healthy scratches. … Defender Alexei Emelin, who has yet to play this season, is expected to dress over the weekend. Emelin underwent knee surgery last May. … The Lightning are first in the Atlantic division. … Montreal heads to Columbus to play the Blue Jackets on Friday.