[2] He also made two appearances in the club's senior team in the Swedish second division league HockeyAllsvenskan, during the 2007–08 season.

Claesson made his Elitserien debut in the Djurgården's first league game of the season on 16 September 2010 against HV71.

[8] Claesson made his NHL debut on 30 December 2015 at Ottawa versus the New Jersey Devils.

[11] He began the 2019–20 season, with the Hurricanes AHL affiliate and reigning champions, the Charlotte Checkers.

Following 10 seasons in North America, Claesson opted to leave as a free agent and was signed to a one-year contract with reigning Russian champions, CSKA Moscow of the KHL, on 13 July 2022.