Tampa Bay had been 0-7 in its previous seven overtime games in the Stanley Cup Playoffs.
"The guys played hard, they are a determined group," Lightning coach Jon Cooper said. "When there are obstacles in the way, they'll try and get through it any way possible."
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. Cirelli had two assists, and 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.
"I felt like every time we were jumping on the ice with a fresh line they always started defending," Montreal coach Martin St. Louis said. "We defended in the whole third (period). I thought we did a really good job in the first two periods playing with the puck and having poise, and it just got away from us."
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).
"Let's be honest, going up there and being down 0-2, your margin of error keeps getting smaller and smaller," Cooper said. "At least we kept our margin of error the same, if not even given ourselves a little bit better of a chance."
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.
"Everyone in the room knows we're good when we play with emotion," Hagel said. "Not just myself, but guys who want to bring everyone into the fight, and that's what I was trying to do a little bit. Glad to get the win, but it's a good team over there and it's going to be like this the entire series, so I'm excited to get to Montreal."