[4] He made his Eliteserien debut for Vålerenga on 30 June 2019 against Haugesund, following injury to first choice goalkeeper Adam Larsen Kwarasey.
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.
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.