Point (two goals, three assists) and Kucherov (one goal, four assists) are the first players in Lightning history to have five points in a postseason game.
Yanni Gourde scored twice, Victor Hedman had a goal and two assists, and Andrei Vasilevskiy made 23 saves for Tampa Bay, the No. 2 seed in the East, which played for the first time since Aug. 31.
"Things are going well right now, but it's our team," Point said. "We're playing so well. We're playing with structure, we're playing responsible with the puck, we're making the right plays. I think our team has been so good."
Thomas Greiss allowed three goals on nine shots for New York, the No. 6 seed, before being pulled at 10:46 of the first period. He was replaced by Semyon Varlamov, who made 20 saves.
"This game, we'll wash it away," Islanders coach Barry Trotz said. "We've lost some heartbreakers in overtime, some real hard matches and all that. Sometimes you get it handed to you like we did today."
WATCH: [All Islanders vs. Lightning Game 1 highlights]
Game 2 is in Edmonton, the hub city for the conference finals and the Stanley Cup Final, on Wednesday (8 p.m. ET; NBCSN, CBC, TVAS).
The team that wins Game 1 is 488-221 (68.8 percent) winning a best-of-7 Stanley Cup Playoff series, including 10-2 in the first two rounds this season.
"[The Islanders are] a proud group," Lightning coach Jon Cooper said. "I don't expect them to roll over for us. It's just one game."
Point gave Tampa Bay a 1-0 lead 1:14 into the first period, driving across the crease and shooting past the outstretched pad of Greiss.
"We haven't played for a week, and you were wondering how the start was going to be," Hedman said. "And then [Point] went out and did that. He set the tone from the start, along with [Kucherov] and [Ondrej Palat] once again. They were unbelievable. But Brayden Point is taking this to another level, that's for sure."
Jordan Eberle scored a power-play goal to tie it 1-1 at 4:33.
"We know how they play," Eberle said. "It was just one of those games. They get a chance and we made a mistake. They get a chance, it ends up in the back of our net."
Hedman gave the Lightning a 2-1 lead with a power-play goal at 8:12 of the first, and Ryan McDonagh made it 3-1 at 10:46 with a slap shot through a screen.