Anthony Cirelli won a face-off back to Moser, who skated across the blue line and down to the top of the right circle before roofing a shot over the glove of Jakub Dobes.
Tampa Bay had been 0-7 in its previous seven overtime games in the Stanley Cup Playoffs.
Brandon Hagel had a goal and an assist, and Nikita Kucherov also scored for the Lightning, who are the No. 2 seed from the Atlantic Division. Andrei Vasilevskiy made 25 saves.
Lane Hutson and Josh Anderson scored for the Canadiens, who are the No. 3 seed from the Atlantic. Dobes made 31 saves.
The best-of-7 series is tied 1-1. Game 3 will be at Bell Centre in Montreal on Friday (7 p.m. ET; CBC, TVAS, SN, TNT, truTV, HBO MAX, The Spot).
Hagel gave Tampa Bay a 1-0 lead at 8:40 of the first period. His initial shot from the slot was blocked by Kaiden Guhle, who was playing without a stick, but the Lightning forward recovered his own rebound at the top of the left circle and beat Dobes with a slap shot to the far post.
Hutson tied it 1-1 with his first career playoff goal at 16:11. He scored with a one-timer from the point that went under Vasilevskiy's blocker arm while on a power play.
Anderson gave Montreal a 2-1 lead at 18:36 of the second period after Jake Evans won a battle along the end boards. Phillip Danault picked up the puck near the left post and sent a short backhand feed out front to Anderson, who chipped it over Vasilevskiy's glove.
Kucherov tied it 2-2 at 12:33 of the third period. Hagel's shot from the high slot went wide of the net, but Kucherov took the puck off the end boards and tucked a wraparound inside the right post before Dobes could get across.
