Kucherov, who had an assist on Yanni Gourde's goal to make it 2-0 at 3:41 of the third period, has five goals and eight assists during a four-game point streak and leads the NHL with 94 points (27 goals, 67 assists).
Carey Price made 34 saves for the Canadiens (31-20-7), who lead the Carolina Hurricanes by one point for the first wild card into the Stanley Cup Playoffs from the Eastern Conference.
"I thought we played a very good game up until [the power-play goal]," Price said. "We laid an absolute egg in the third period. It's a frustrating result, for sure, because I thought we played very well for 40 minutes."
Kucherov put the Lightning ahead 1-0 with a power-play goal at 2:18 of the third period. He scored with a one-timer from the right face-off circle off a cross-ice pass from Steven Stamkos.
"It's big. When it's 0-0 going into the third, the next one is going to be a big one, and it was," Stamkos said. "It was a big-time player that came through, and [Vasilevskiy] was great again."