On 29 May 2012, the Boston Bruins announced that they had signed Svedberg from the Swedish Elite League to an Entry Level Contract.

[4] After leading all KHL goaltenders in games won during 2015–16, he re-signed a two-year contract extension with Salavat.

On June 8, 2018, Svedberg as an impending free agent signed a two-year contract to return to Sweden with Timrå IK in the SHL.

[6][7] In the 2018–19 season, Svedberg registered just 10 wins in 35 games with Timrå IK, unable to keep the club from suffering relegation to the Allsvenskan.

In order to continue in the SHL, Svedberg left Timrå IK and agreed to join original hometown junior club, Djurgårdens IF for the final year of his contract on 7 June 2019.