After taking a pass from Barclay Goodrow from below the goal line, Cirelli shot off the left post before it hit the back of Semyon Varlamov's pad and crossed the goal line.
"The emotion is just so high," Cirelli said. "We worked all year. Our goal was to be playing for the Stanley Cup. We're here now, so I think it's every kid's dream to be in this situation."
Brock Nelson had a chance to win it for the Islanders while they were killing a four-minute power play. He stole the puck from Tyler Johnson and skated in on a breakaway, but Andrei Vasilevskiy made a shoulder save at 2:24 of overtime.
WATCH: [All Lightning vs. Islanders Highlights]
Victor Hedman scored, and Vasilevskiy made 26 saves for Tampa Bay, the No. 2 seed, which will play the Dallas Stars in the Cup Final.
Game 1 of that best-of-7 series will be in Edmonton, the hub city, on Saturday (7:30 p.m. ET; NBC, CBC, SN, TVAS).
"The exuberance and relief, it's unparalleled how you feel inside," Lightning coach Jon Cooper said. "It's about the players and the effort, dedication and perseverance they've put into this. They're the ones that deserve this because they've done all the work."