Mark Streit, with another short-handed goal, and Maxim Lapierre also scored while Sheldon Souray had three asissts for the Canadiens (18-8-5), who have collected at least a point in five straight games (3-0-2). Brad Richards and Martin St. Louis scored for the Lightning (14-15-2), who have won only once in their last eight games.
The loss was Tampa's third straight, and all of them have been by a single goal.
Cristobal Huet made 41 saves for the Canadiens to run his record to 12-3-3, keeping Montreal in the game despite being completely dominated through the first 40 minutes.
Marc Denis made 16 saves for the Lightning and dropped to 6-10-2.
Streit and Koivu's short-handed markers gave the Canadiens a league-leading total of nine on the
Vincent Lecavalier extended his point streak to eight games with an assist on St. Louis' goal in the second period. Lecavalier has eight goals and seven assists during the streak, while St. Louis has eight goals and 13 assists in his last 12 games.
With the Lightning ahead 2-1 in the third, Lapierre tied the game with his first career goal when he tipped in Mike Komisarek's point shot at 6:38.
Craig Rivet took a tripping penalty at 12:06 of the third, but it was the Canadiens who capitalized to take the lead. Alex Kovalev swept into the Tampa zone on a 2-on-2 rush before dropping a pass for Koivu, whose wrist shot found the far top corner for his 11th of the season and a 3-2 Montreal lead.
Tampa's Eric Perrin looked to have tied it with 3:27 left in regulation, but a lengthy video review ruled that the puck never crossed the goal line despite the fact the red light came on.
Koivu iced it when he outraced Tampa defenceman Paul Ranger to score into an empty net at 19:14 of the third for his 12th of the season.
The Canadiens were lucky to be going into the third period down only 2-1 as the Lightning outshot them 33-10 in the first two periods and completely dominated play. The tight score was largely thanks to Huet, who made highlight-reel stops on Lecavalier in the first and Filip Kuba midway through the second.
Montreal scored on its first shot of the game at 8:29 of the first when Souray fed Streit with a perfect pass on a shorthanded 2-on-1 break for the Swiss defenceman's first goal of the season.
The Lightning tied it on the power play at 13:39 of the first when Richards converted a Vaclav Prospal feed with a falling one-timer for his 13th of the season.
Tampa Bay took the lead 2-1 at 18:42 of the second when, on a delayed penalty, Lecavalier's shot was stopped by Huet but the rebound went straight to St. Louis on the other side of the net, and he buried it for his 19th of the year.
Notes: Canadiens captain Saku Koivu was playing in his 600th career game, while centre Tomas Plekanec played in the 100th of his career. . . . The Montreal Canadiens unveiled "Builders Row" in sections 104 through 107 of the Bell Centre earlier Thursday as part of the team's build-up to its 100th anniversary in 2009. The initial inductees were Canadiens founder John Ambrose O'Brien, special advisor William Northey, former team president Donat Raymond and former owners Leo Dandurand, Jos Cattarinich, Louis Letourneau and Senator Hartland de Montarville Molson. In 2007, the Canadiens will add former general managers, and in 2008 former coaches will be added.