Slafkovsky beat Andrei Vasilevkiy with a snap shot from the left face-off circle after Tampa Bay forward Jake Guentzel was given a high-sticking minor with 21 seconds left in regulation.
Josh Anderson scored, and Nick Suzuki and Cole Caufield each had two assists for the Canadiens, who are the No. 3 seed from the Atlantic Division. Jakub Dobes made 20 saves.
Brandon Hagel scored twice, Darren Raddysh had a goal, and Guentzel had three assists for Lightning, who are the No. 2 seed from the Atlantic. Nikita Kucherov had two assists, and Vasilevskiy made 15 saves.
Tampa Bay is 0-7 in its past seven playoff overtime games.
Game 2 is here on Tuesday (7 p.m. ET; ESPN2, SN, CBC, TVAS, The Spot)
Anderson put the Canadiens up 1-0 at 13:24 of the first period. Alexandre Carrier won a puck battle below the goal line and sent the puck through the legs of Lightning forward Gage Goncalves out in front to Anderson, who snapped a shot past Vasilevskiy's glove.
Anderson appeared to make it 2-0 at 10:43 of the second period when he tipped Mike Matheson's left point shot from in front, but the goal was overturned after video review determined Anderson hit the puck above the crossbar.
Raddysh tied it 1-1 with a power-play goal at 12:15. Kucherov fed a pass to Raddysh at the top of the left circle, where he put a one-timer by Dobes' blocker.
Hagel gave Tampa Bay a 2-1 lead just 29 seconds later at 12:44, pouncing on a loose puck at the left of the net and lifting a backhander to the glove side from in tight.
Slafkovsky evened the score 2-2 while on the power play at 19:36 with a one-timer from the right circle that went over the outstretched glove of Vasilevskiy.
Slafkovsky put Montreal up 3-2 with his second power-play goal at 5:56, taking a pass from Caufield and sneaking a snap shot from the slot under Vasilevskiy's left pad.
Hagel tied it 3-3 at 8:58 when he tapped in a backdoor pass from Guentzel from atop the crease into an open net for his second goal of the game.
