Steven Stamkos scored his third goal of the playoffs for the Lightning, who closed out the second-round series in six games after Montreal fought off elimination in Games 4 and 5.
"This was our Game 7 tonight," Stamkos said. "This is the biggest game of the playoffs for us and we responded."
Kucherov finished with six goals in the series.
Ondrej Palat also had a goal and assist for Tampa Bay, which played without forward Ryan Callahan, who had an emergency appendectomy on Monday night.
Ben Bishop stopped 18 shots to outperform Carey Price, who finished with 24 saves.
The Lightning, who won nine of 11 meetings against the Canadians this season including playoffs, will face either the New York Rangers or Washington Capitals in the conference finals.
Those teams meet in Game 7 of their second-round series on Wednesday night.
Price, a leading candidate for league MVP, played well in helping the Canadiens climb back in the series after losing the first two games at home and watching Game 3 slip away in the closing seconds for a 3-0 deficit.
The Montreal star stopped several shots from point-blank range — but Bishop was better for Tampa Bay.
Montreal's loss means the last of five Canadian-based teams that reached the postseason is eliminated from the playoffs. A Canadian-based team hasn't won the Cup since the Canadiens did it in 1993.
Kucherov tipped Ondrej Palat's shot from the right circle over Price's left shoulder to get the Lightning started at 15:35 of the opening period.
Stamkos, who urged his teammates to not play "scared" in trying to end the series and avoid a trip to Montreal for Game 7, scored from the slot to make it 2-0 early in the second.
Palat scored on the power play just over a minute before the second intermission.
Bishop stopped Dale Weise on a breakaway with just under 13 minutes to go. He was less than five minutes away from a shutout when Max Pacioretty broke through for Montreal with his fifth goal of the playoffs.
Kucherov finished the scoring with an empty-net goal.
The Canadiens won Game 4 in Tampa to avoid a sweep, then took Game 5 at home on P.A. Parenteau's goal with just over four minutes remaining to send the series back to Florida with the Lightning clinging to a 3-2 lead.
The teams shrugged off questions about who was feeling more pressure to win, with the Canadiens saying Tampa Bay had to be feeling the heat to close after losing two straight.
And the Lightning insisting there was no need to panic because they still held the series edge and were back at Amalie Arena, where they compiled the league's best home record during the regular season.
Bishop was on top of his game from the start, making a nice glove save on Tomas Plekanec's point-blank shot a little over 13 minutes into the game to keep Montreal from scoring first, and then handling everything the Canadiens threw at him after Tampa Bay went ahead.
Notes: Montreal outshot the Lightning in each of the first five games, but held a 28-19 advantage Tuesday night. ... Despite the loss, the Canadians have won five of the last seven games they've played when facing elimination. ... Lightning rookie forward Jonathan Drouin, one of the options coach Jon Cooper considered for replacing Callahan, was a healthy scratch.
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