The Canadiens looked strong, going up 2-0, but after that it became a shaky evening.
A late penalty to PK Subban proved costly, setting up the tying goal for Tampa late in the third period.
Tomas Plekanec had a goal and an assist for the Habs in a losing effort.
The opening ceremony featured the 2010-11 Habs lined up on the faceoff circle. The players were then introduced by captain Brian Gionta, in French, that rallied the Bell Centre crowd.
Maxim Lapierre opened the scoring, after some hard work in the corners by Lars Eller forced a Tampa turnover and found the Habs winger in front of goalie Mike Smith at 5:22
Playing four-on-four, Mike Cammalleri found his own rebound, and fooled Smith with a pass to Plekanec for his first goal of the season at 18:07.
"We were soft, we were tentative, we were hesitant," said Lightning coach Guy Boucher on his team's first period performance, despite outshooting Montreal 18-13. "I didn't expect us to freeze out there."
Tampa got on the board on the power play, at 1:44 of the second period, when Brett Clark scored his goal on a slap shot that beat a heavily screened Price.
The Lightning then tied the game briefly at 10:54 of the third period, when Martin St. Louis found a Steven Stamkos rebound to beat a helpless Price.
Thirty-one seconds later Plekanec fed Andrei Kostitsyn, who made a slick deke on Smith to regain the lead.
With just over two minutes remaining, Subban took a costly slashing call that he could have avoided. The Lightning capitalized and with their goalie pulled, Stamkos forced overtime with 1:19 to play.
The Bell Centre fans, outraged that a penalty was not called for an infraction against Dustin Boyd, littered the ice in disgust, causing a brief delay.
Through the three periods, Price made 42 saves, many that were game savers.
"He (Price) was outstanding. He was the one that's given us our last three points," Plekanec said of his goalie. "In the second and third period, we just didn't execute."
In overtime, Malone came around the back of the net and chipped in a loose puck past a crowded Montreal crease to take the win.
"It's not about me, it's about the team," said a dejected Price. "It's not what we wanted. We took too many penalties and we can't do that."
The Canadiens now travel to Buffalo, for a Friday evening game with the Sabres, before returning home Saturday to face Ottawa.
What the Tampa fans are saying about the game at Raw Charge
MTL - Maxim Lapierre (unassisted) 5:22
MTL- Tomas Plekanec (Cammalleri, Spacek) 18:07
TB- Brett Clark (PP) (unassisted) 1:44
TB- St. Louis (Stamkos) 10:54
MTL- Kostitsyn (Plekanec, Gorges) 11:25
TB- Stamkos PP (Lecavalier) 18:41
TB -Malone (Moore) 4:29
Three Stars 1. Malone 2. Price 3. Plekanec