Mathew Barzal scored for the Islanders, who have lost three in a row. Semyon Varlamov stopped 24 shots.
It was the Islanders’ 12th of 13 games during a season-opening road trip. New York will play its home opener at the new UBS Arena on Saturday night against Calgary.
The teams met for the first time since Tampa Bay beat the Islanders 1-0 in Game 7 of the Stanley Cup semifinals last June.
New York defenseman Zdeno Chara and Lightning forward Pat Maroon fought right after the opening puck drop. Chara was given a high-sticking minor and 10-minute misconduct, and Tampa Bay rookie Boris Katchouk received a 10-minute misconduct following a scrum late in the third.
Stamkos gave Tampa Bay a 4-1 lead during the ensuing power play with 4 1/2 minutes remaining. Point had made it 3-1 with 6:32 to play.
Vasilevskiy had several in-close saves, including one on Anders Lee during a second-period power play.
Barzal opened the scoring 2:53 into the first on a breakaway, but Joseph tied from the low slot 23 seconds later.
Cirelli put the Lightning up 2-1 shortly after a power play ended with 37.3 seconds left in the first.
There were several scrums and a fight between Cirelli and Brock Nelson during the second. Nelson received the initial penalty for charging, while Cirelli was given an instigating minor and 10-minute misconduct.
Cirelli departed shortly after returning in the third after it appeared he was hit in the face by deflected shot.
New York failed to score on three power plays and are 4 for 31 overall.
Islanders: D Andy Greene became the 358th player to play in 1,000 games. … Chara is the 11th player to appear in 1,620 games. … Ryan Pulock played in his 263rd consecutive game, breaking Denis Potvin’s team record for defensemen. Pulock left after taking a shot off his leg in the second but returned.
Islanders: C Casey Cizikas (illness) didn’t play.
Lightning: D Erik Cernak (upper body) missed his second straight game but is skating.
Islanders: At Florida on Tuesday night in the finale of their season-opening stretch of road games.
Lightning: At Philadelphia on Thursday night.