James Wright also scored for Tampa Bay, which went 2-0-1 after dropping the opener of a four-game road trip at Philadelphia on Monday.
Antero Niittymaki made 36 saves in his third straight start for the Lightning (6-4-5).
Brian Gionta scored his seventh goal for Montreal (8-9-0), which has lost four of six.
Carey Price, who stopped 18 shots, fell to 3-7.
Malone opened the scoring with his 10th goal in 15 games on a power play at 10:09.
Lightning captain Vincent Lecavalier recorded his second straight assist when Tanguay's third goal of the season put Tampa Bay up 2-0 with 46.3 seconds left in the first.
Tanguay, who signed a one-year deal with the Lightning after he failed to receive an offer from Montreal, scored for the second straight game when he flipped a rebound of Malone's shot from the left side past Price for his third goal.
Gionta drew the Canadiens within 2-1 with a power-play goal midway through the second.
The Lightning regained their two-goal margin after Wright got credit for his second of the season at 18:02.
A video review confirmed referee Dennis LaRue's understated ruling that the puck had crossed the line before Montreal's net became dislodged during a goalmouth scramble.
Limited to one point in his first seven games with Tampa Bay, Tanguay earned an assist on Wright's goal for his seventh point in six games.
Notes
Rookie Lightning D Victor Hedman didn't play. Hedman, chosen second overall by Tampa Bay in this year's draft, sat out for precautionary reasons after a hit by Ottawa's Chris Neil knocked him out of Thursday's 3-2 overtime loss to the Senators. … Montreal C Kyle Chipchura and LW Gregory Stewart were healthy scratches for a second straight game in favor of C Tom Pyatt and RW Ryan White, who made their NHL debuts Thursday. … D Jay Leach, picked up by Montreal off waivers from New Jersey on Friday, didn't play.