The Canadiens moved within three points of Boston for the lead in the Northeast Division and three points behind Tampa Bay in the battle for fifth place in the Eastern Conference.
After Simon Gagne scored for Tampa Bay in the third round of the shootout to tie it, Cammalleri put a wrist shot between the legs of Dwayne Roloson. Carey Price followed with a save on Teddy Purcell's shot to give the Canadiens their eighth win in 11 games.
It was a 1-1 duel heading into the third period before Brian Gionta gave Montreal the lead with a shorthanded goal 3:58 into the stanza. Gagne forced overtime by scoring a power-play goal midway through the third.
Tampa Bay has won just twice in its last nine games.
Price made 35 saves to grab the win. Roloson allowed two goals on 29 shots.