Cirelli played a key role for the Tampa Bay Lightning in their run to the Stanley Cup championship in 2020. His overtime goal in Game 6 of the Eastern Conference Final against the New York Islanders put the Lightning into the Stanley Cup Final for the first time since 2015, and he contributed three assists in six games against the Dallas Stars in the Cup Final to help Tampa Bay win its first championship since 2004. His 12 points (five goals, seven assists) in 23 games of the 2021 Stanley Cup Playoffs helped the Lightning repeat as champions.

During the 2024-25 season, part of a stretch when Cirelli was playing some of the best hockey of his career, he was named to Canada's roster for the 4 Nations Face-Off. He finished that season with NHL career highs of 27 goals, 32 assists and 59 points.

Cirelli was named to Team Canada's roster for the Olympic Winter Games Milano Cortina 2026 but replaced by Florida Panthers forward Sam Bennett because of an injury. He was named a finalist for the 2026 Selke Trophy, awarded annually to the forward voted best to excel in the defensive aspects of the game, for the second straight season. Cirelli had 52 points (23 goals, 29 assists) in 71 games was an NHL career-best plus-38 and led the Lightning with 1,075 face-offs, winning 48.2 percent.

The Lightning selected Cirelli in the third round (No. 72) of the 2015 NHL Draft after he had 36 points (13 goals, 23 assists) for Oshawa of the Ontario Hockey League. He scored twice including in overtime in the 2015 Memorial Cup final against Kelowna of the Western Hockey League, and signed his entry-level contract with Tampa Bay on May 19, 2016.

In 2016-17, he had 34 points (13 goals, 21 assists) in 26 games for Oshawa before he was traded to Erie on Jan. 9, 2017. He had 30 points (12 goals, 18 assists) in 25 games with Erie, which advanced to the 2017 Memorial Cup Final, losing 4-3 to Windsor of the OHL. He began the 2017-18 season with Syracuse of the American Hockey League and had 37 points (14 goals, 23 assists) in 51 games before being recalled on March 1, 2018. He made his NHL debut that night against the Dallas Stars and scored his first goal and assist in a 5-4 overtime win.

Cirelli took a big step forward in 2018-19, settling in as a third-line center and finishing with 19 goals, 39 points and a plus-25 rating to help the Lightning tie the NHL single-season record with 62 victories. He continued to boost his offensive numbers in 2019-20, scoring 16 goals and finishing with 44 points and a plus-28 rating.