Steven Stamkos and Tyler Johnson each scored two goals for the Lightning (28-7-2), who are 11-0-1 in their past 12 games, the longest point streak in the NHL this season. Andrei Vasilevskiy made 25 saves.
It was Kucherov's first five-point game in the NHL.
"It's a tough building to play in, they're a good team, but I thought we played 60 minutes of really good hockey," Kucherov said.
WATCH: [All Lightning vs. Oilers highlights]
Connor McDavid had three assists, and Ryan Nugent-Hopkins, Alex Chiasson and Leon Draisaitl scored for the Oilers (18-15-3), who have lost three games in a row. Mikko Koskinen made 39 saves.
"We can play with a team like that," Draisaitl said. "If we take out a few errors, team-wise, structure-wise and individually, we saw for a big part of the game that we can play with a team like that. That's a good sign, but they scored three more goals than we did."
Nugent-Hopkins gave Edmonton a 1-0 lead at 10:15 of the first period when he scored off a pass from McDavid.
It was McDavid's 200th NHL assist. He is the sixth player in NHL history to reach 200 before turning 22 (Wayne Gretzky, Sidney Crosby, Mario Lemieux, Dale Hawerchuk, Ron Francis).