Reilly Smith and Alec Martinez scored for the Golden Knights, who tied the Colorado Avalanche for the most points in the NHL in the regular season (82), losing the Presidents' Trophy on the regulation-wins tiebreaker (35-30). Vegas lost in the Stanley Cup Final in 2017-18 and in the Western Conference Final last season.
The Golden Knights went 0-for-2 on the power play and were 0-for-15 in the series.
"When you get to this point, the teams that win find another level," Vegas coach Peter DeBoer said. "They don't sag. We didn't find another level, and they did. I thought they were opportunistic. They owned the key moments of the series.
"When they got a chance, they stuck it in the net. When they needed a big save, they got a big save. They won the overtime battle. They won the special-teams battle. If you're losing those areas of the game, you're putting yourself in a tough spot."
Lehner made 29 saves in his second start of the series after Marc-Andre Fleury allowed three goals on 25 shots in Game 5, a 4-1 loss.
"It's hard to take in, but I thought we battled hard," Lehner said. "I'm really proud of our guys' season. Everyone worked really hard, and it's not easy to get to the Stanley Cup Final. We were right there."
Canadiens captain Shea Weber made it 1-0 with a power-play goal at 14:06 of the first period. Martinez's backhand on a clearing attempt went straight to Weber, who scored on a slap shot from the top of the left face-off circle.
"I'm definitely proud of everybody in that locker room and what we've accomplished so far, but there's still work to be done," Weber said.