Claesson started his career with local clubIFK Värnamo. After progressing through the youth ranks, he made his senior debut in 2009 in aDivision 1 Södra match. After a breakthrough campaign during the2011 Superettan season, Claesson joinIF Elfsborg in theAllsvenskan, the top division of Swedish football.
Out of action for over 14 months, Claesson returned from injury on 18 August 2020, playing the last 15 minutes of Krasnodar's 2–0 home win overArsenal Tula in theRussian Premier League.[8] On 18 September, he scored his first two goals since his return from injury in a 7–2 thrashing ofKhimki.[9] Four days later, Claesson scored from the penalty spot in the club's first-legplay-off round match againstPAOK for qualification to the group stage of theUEFA Champions League.[10] Krasnodar would win the second leg against PAOK by the same scoreline and secure progression to the group stage of the Champions League for the first time in the club's history.[11]
On 3 March 2022 Krasnodar announced that his contract was suspended and he would not train with the team, but the contract was not terminated and remained valid.[12] On 5 March 2022, his contract was terminated and he was released from the club.[13]
On 30 March 2022, Claesson signed withCopenhagen in Denmark until the end of the season.[14] On 3 April, he scored the winning goal in his debut match for the club after coming on as a substitute in the 61st minute.
On 16 June 2022, Claesson signed an extension with Copenhagen until 2026.[15]
In May 2018, Claesson was named inSweden's 23-man squad for the2018 FIFA World Cup in Russia.[16] He was one of Sweden's main men for the duration of the tournament, in which the Blågult reached the quarterfinals.
In aUEFA Euro 2020 qualifier againstSpain on 10 June 2019, Claesson suffered a serious knee injury, rupturing the cruciate knee ligaments, after a clash withJordi Alba.[8]
He was named in Sweden'sUEFA Euro 2020 squad, and appeared in all four games as Sweden was eliminated in theround of 16 byUkraine.[17] He scored the winning goal in a 3–2 win againstPoland in the group stage.[17] On 19 November 2023, Claesson registered one goal and one assist against Estonia in their 2–0 win in their European Qualifiers game.
In October 2025, he announced his retirement from the national team. He was capped 74 times and scored 15 goals.[18]