Jesse Ylonen put the Canadiens ahead 3-2 with a power-play goal at 12:23 for his third goal in four games.
Steven Stamkos scored his 30th goal of the season to tie it 3-3 on the power play at 3:14 of the third period, a one-timer off a pass from Kucherov.
"We stepped on the gas [in the third period]," Hedman said. "These are tough games. There's not an easy game in this league and we knew that coming in. [The Canadiens] have lost games, but they've scored a lot of goals in those losses."
Canadiens forward Jonathan Drouin dressed but did not get a shift after St. Louis said he would be disciplined for missing a team meeting Friday.
"I think tonight was a fair thing, and 'Jo' handled it tremendously well," St. Louis said. "Me and Jo have a great relationship. I got to know him a lot this year, and he's had a great season for us this year. I love the kid, and for him to handle it the way he did, I gained even more respect for him."
NOTES: St. Louis was part of the inaugural class inducted into the Lightning Hall of Fame this weekend. He was joined by forward Vincent Lecavalier and team founder Phil Esposito. … Stamkos reached 30 goals for the eighth time in the NHL. … Hagel played his 200th NHL game. His first hat trick came in his 100th game, an 8-5 win against the Devils on Feb. 25, 2022, when he was with the Blackhawks. … Stamkos (30 goals, 40 assists) reached 70 points for the eighth time, tying St. Louis for the most in Lightning history. … Kucherov became the fourth player in Lightning history with 200 multipoint games, joining Stamkos (281), St. Louis (257) and Lecavalier (212). … Sergachev has 11 assists in the past seven games. He is the third Lightning defenseman to reach 45 assists in a season (Hedman, three times; Roman Hamrlik, once). … Anderson has five points (two goals, three assists) in a four-game point streak.