Josh Anderson scored twice for the Canadiens, including the game-winner 3:27 into overtime. Nick Suzuki had the other goal while Carey Price made 32 saves.
Barclay Goodrow and Pat Maroon lit the lamp for Tampa Bay with Andrei Vasilevskiy stopping 18 shots. The Lightning will have a second chance to clinch the Cup in Game 5.
Anderson opens, closes scoring as Habs edge Lightning
The opening half of the first period belonged to Tampa Bay as they dominated the action at both ends of the ice, peppering Price with shots as they had the first eight shots on goal. Despite that, Montreal struck first.
Suzuki played the puck to Cole Caufield, who immediately fed it back to Suzuki as he worked below the goal line. Suzuki flicked a puck back to the front of the net where Caufield caught a piece of it, but it was Anderson who roofed the chance to give Montreal a 1-0 lead with 4:21 left in the period.
Jake Evans slowed up Brayden Point and in return, Point punched Evans in the face three times, and the officials sent both to the box on coincidental minors. That four-on-four situation soon became a four-on-three as Joel Edmundson was whistled for slashing in the neutral zone. The Lightning power play had a great chance as Point exited the penalty box and rang his shot off the crossbar.
Yanni Gourde and Edmundson exchanged shoves near the end of the period, eventually, Anderson tracking down Pat Maroon with both players getting penalized for roughing, resulting in the teams starting four-on-four in the second period began.
Montreal was handed a power play after Point was called for high-sticking on Arturri Lehkonen. Vasilevskiy stood tall in the Tampa Bay net and helped the champions kill off the man advantage. As he did in the first period, Point tried to create a scoring chance coming out of the box, but Shea Weber hit him along the boards to stop that.
Corey Perry then was whistled for grabbing a Lightning jersey and Joel Armia called for tripping. Both penalties were killed off, but not before Victor Hedman rang one off the post after the puck caught the tip of Price's blocker.
More sloppy turnovers led to the Lightning tying the game as a clearing pass from Jeff Petry was intercepted by Ryan McDonagh and as the puck was passed back through the Montreal zone, he threw the final pass back into the slot for Goodrow to finish off and tie the game 1-1.
Withe the score even with 20 minutes to play, Tampa Bay applied severe pressure, eventually leading to a pileup around Price, which in turn led to three Lightning and three Canadiens players taking a very long seat in the penalty box.
With Tampa Bay's entire third line in the box and Montreal down two of their top four defenders, an unlikely hero emerged as Romanov, a healthy scratch all series, took a pass from Jake Evans and rifled a shot past Vasilevskiy to make it 2-1 Canadiens with 11:12 left.
One failed dump-in by Tyler Toffoli instantly turned into a two-on-one for Tampa Bay, where Mathieu Joseph fed Maroon, who tied the game with 6:12 remaining in regulation.
A double-minor by Weber for high-sticking that carried into overtime gave the Lightning a chance to win the game, but Montreal escaped after killing off the first minute in the third period and the rest to begin overtime.
Phillip Danault and Suzuki got a two-on-one rush, but a save from Vasilevskiy denied them. However, it was Montreal who did come up with the winner as Anderson raced after the puck after he had poked it free, sending it in front of the net. Caufield held up McDonagh, created a rebound chance and Anderson swatted the rebound home to keep the Canadiens season alive.