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.
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.