Klaesson signed his first professional contract in June 2016.[4] He made his Eliteserien debut for Vålerenga on 30 June 2019 against Haugesund, following injury to first choice goalkeeper Adam Larsen Kwarasey. The game ended with a 4–1 win for Vålerenga. Klaesson also started the following game, a 6–0 win against Bodø/Glimt.
Leeds United
On 31 July 2021, Klaesson signed a four-year deal with Leeds United, with the club paying an undisclosed fee reported as £1.6 million.[5][6]
He made his Premier League debut on 18 March 2022 in a 3–2 victory away to Wolverhampton Wanderers, following an injury to first choice goalkeeper Illan Meslier. During his time on the pitch, he did not concede any goals.
Klaesson made his first start for the club on 1 January 2024 at home against Birmingham City. He played the full 90 minutes, keeping a clean sheet in the 3–0 win.
Raków Częstochowa
On 7 July 2024, Klaesson moved on a free transfer to Ekstraklasa club Raków Częstochowa on a three-year contract.[7] Having arrived at the club out of shape,[8] he left Raków by mutual consent on 26 August that year without making a single first-team appearance.[9] According to reports from Polish media outlets, such as Weszło, his exit was allegedly due to concerns over his physical condition, with speculation arising after fans reportedly spotted him at a McDonald's.[10][11]
AGF
On 30 September 2024, Klaesson signed a contract with Danish Superliga club AGF until July 2025.[12] He was set to replace backup Leopold Wahlstedt, who had broken his ankle in a match a few days earlier.[13]