Kucherov, who turned 26 on Monday, also won the Ted Lindsay Award as the league's outstanding player as selected by his fellow players.
A year after scoring 100 points, Kucherov emerged as a dominant NHL forward for the powerhouse Lightning. He won the Art Ross Trophy for leading the league in scoring with a career-best 41 goals and 87 assists.
He tied Jaromir Jagr's NHL record for assists by a wing, and he surpassed Alexander Mogilny's single-season record for points by a Russian-born player. His 128 points were the most by any NHL player since 1996 -- and the Lightning coincidentally also had 128 points in the standings, also the most by any team since 1996.
Earlier in the night, Lightning teammate Andrei Vasilevskiy took home his first Vezina Trophy as the NHL's top goalie.
Vasilevskiy led the NHL with 39 victories for the powerhouse Lightning, who took the Presidents' Trophy with 128 points. The Russian goalie had a 2.40 goals-against average and a .925 save percentage in 53 games.
Vasilevskiy is the first Lightning goalie to win the Vezina.
Canucks center Elias Pettersson opened the NHL Awards show by winning the Calder Trophy as the NHL's best rookie.
The 20-year-old Pettersson beat out St. Louis goalie Jordan Binnington and Buffalo defenseman Rasmus Dahlin for the award. Pettersson joins Pavel Bure (1992) as the only Calder winners in Canucks history.
Pettersson provided an offensive jolt to the Canucks in his first North American season, scoring a goal on his first NHL shot and seamlessly adapting to the world's best league. One year after Vancouver drafted him with the fifth overall pick in 2017, Pettersson posted a rookie-best 28 goals and 38 assists as a tremendous playmaker with a wicked shot.
Pettersson is the 31st player in NHL history to lead all rookies in goals, assists and points.
Calgary's Mark Giordano won the Norris Trophy as the NHL's top defenseman.
The 35-year-old Giordano was awarded the Norris for the first time in a 13-year NHL career spent entirely with the Flames. Giordano beat out two vaunted finalists: San Jose's Brent Burns, who won it in 2017, and Tampa Bay's Victor Hedman, who won it last year.
The Calgary captain was instrumental in the Flames' breakthrough season, leading the NHL with a plus-39 rating while regularly playing against opponents' best lines. Giordano also scored a career-best 74 points and played more than 24 minutes per game.
Ryan O'Reilly of the Stanley Cup champion Blues won his first Selke Trophy as the NHL's top defensive forward.
O'Reilly was the Blues' top scorer with 77 points, including 28 goals in his first season in St. Louis. He also compiled a plus-22 rating and finished fourth in the NHL with 94 takeaways while winning an NHL-best 1,086 faceoffs during the Blues' remarkable season.
The Islanders' Barry Trotz won the Jack Adams Award as the league's top coach.
Trotz beat out Tampa Bay's Jon Cooper and St. Louis' Craig Berube for the honor, which is based on regular-season performance. Trotz also won the award with Washington three years ago.
After winning the Stanley Cup and subsequently leaving the Capitals last summer, Trotz engineered an impressive one-season turnaround for the Islanders. New York went 48-27-7 for a 23-point increase from 2018 and its best single-season total since 1983-84, even after losing John Tavares to Toronto.
Boston's Don Sweeney was named the NHL's General Manager of the Year, and Florida center Alexander Barkov won the Lady Byng Trophy as the player best combining sportsmanship, gentlemanly conduct and ability.
Sweeney has built a consistent winner since the longtime Bruins defenseman took over the front office in 2015. Boston reached the Stanley Cup Final this season, losing in seven games to St. Louis.
Barkov scored a franchise-record 96 points for the Panthers, yet he committed just four minor penalties while playing more than 22 minutes per game. The Finn was a Lady Byng finalist for the third time in four seasons.
Barkov drew a few laughs when he apparently heard a few shouts in Finnish as he took the stage to accept his trophy.
"We have more fans from Finland than Florida here,'' Barkov said.
The other Lady Byng finalists were O'Reilly and Calgary's Sean Monahan.
Wild forward Jason Zucker won the King Clancy Memorial Trophy for leadership and humanitarian contributions to hockey. Zucker, who is from Las Vegas, has done extensive fundraising for children's causes in Minnesota.
Lehner was awarded the Masterton Trophy, given to the player exemplifying the qualities of perseverance, sportsmanship and dedication to hockey.
Lehner had an outstanding season after revealing during training camp that he has struggled with addiction and bipolar disorder. He is the third player in Islanders history to win the award.
"I'm not ashamed to say I'm mentally ill, but that doesn't mean mentally weak,'' Lehner said after accepting his award.
The Associated Press contributed to this report.