Jeannot became known as "The Oxbow Ox" when his hometown of Oxbow, Saskatchewan, recognized his physical nature. He played major junior hockey for Moose Jaw in the Western Hockey League and was signed by the Nashville Predators as an undrafted free agent to a three-year, entry-level contract April 2, 2018.

Jeannot played minor hockey in southeast Saskatchewan, his last season with the Estevan Apex midget AA Bruins in 2012-13, when he scored 15 goals and 24 points in 32 games. He then played one season for the midget AAA Yorkton Maulers before joining Moose Jaw, where in his final season he scored 80 points (40 goals, 40 assists) in 72 games.

The performance earned Jeannot his first NHL contract with the Predators. He made his NHL debut in a 4-2 win against the Carolina Hurricanes on March 2, 2021 and finished the regular season with five points in his final seven games. He signed a two-year contract July 30 after scoring 21 points (10 goals, 11 assists) in 13 games for Chicago in the American Hockey League, and six points (three goals, three assists) in five games for Florida of the ECHL.

Jeannot was traded to the Tampa Bay Lightning on Feb. 26, 2023 for defenseman Cal Foote and five draft picks. He signed a two-year, $5.33 million contract with Tampa Bay on July 15 (an average annual value of $2.665 million), and was traded to the Los Angeles Kings on June 29, 2024, for a fourth-round pick in the 2024 NHL Draft and a second-round pick in the 2025 NHL Draft.

He signed a five-year, $17 million contract with the Boston Bruins on July 1, 2025.