Kuba tapped in a perfect pass from Eric Perrin that left goalie Rick DiPietro in a defenceless position. DiPietro made his first start after missing most of three games because of injuries sustained in a mid-ice collision with Montreal's Steve Begin one week earlier.
The loss was the fourth straight on a road trip for the Islanders, who rallied from a 2-0 deficit with three second-period goals against Johan Holmqvist, an embattled goaltender who rebounded from the Lightning's 7-1 loss to Washington on Sunday to beat New York for the fourth time.
Tampa Bay swept the season series from the Islanders 4-0 and has beaten New York eight consecutive times.
The Islanders went 0-3-1 on their trip and dropped to 11th place in the Eastern Conference, one point and three spots below the playoff cutoff in the Eastern Conference.
Perrin scored a power-play goal in the opening period and Lecavalier's second-period goal made it 2-0. Richard Zednik, Richard Park and Miroslav Satan scored for New York before Tampa Bay's Jason Ward scored less than five minutes into the third period to make it 3-3.
Holmqvist stopped 21 shots for the Lightning. DiPietro finished with 36 saves.
The Lightning outshot the Lightning 17-4 in the opening period, but only led 1-0 after going 1-for-4 on the power play. Lecavalier's goal made it 2-0 early in the second, but Zednik countered for New York 14 seconds later to get the Islanders back in the game.
Park's short-handed goal made it 2-2 and Satan put New York ahead with a shot over Holmqvist's left arm with 2:36 remaining in the second period. Ward's first goal for Tampa Bay since being acquired from Los Angeles in a trade last month set the stage for overtime.
Holmqvist and Marc Denis were both pulled during Tampa Bay's embarrassing loss to Washington, another team the Lightning have dominated this season.
Minor league goalie Karri Ramo was recalled from Springfield of the AHL on Monday, and Lightning coach John Tortorella said he wasn't added to the roster to merely watch the team's more experienced goalies.
Notes: Lecavalier has 499 NHL points. He has 15 goals and 15 assists in 31 career games against the Islanders. ... Ward's goal was his first since Jan. 11 when he was with the New York Rangers. ... Tampa Bay was 1-for-6 on the power play. The Islanders were 0-for-2.