On September 28, 2012, Burström as an emerging defenseman signed a four-year contract extension to remain with Champions Skellefteå.[2]
After completing his tenth season within Skellefteå's organization following the 2016–17 season, Burström left as a free agent seeking a new challenge in agreeing to a two-year contract with Russian club, HC Vityaz of the KHL on May 3, 2017.[3] In the 2017–18 season, Burström appeared in 29 games from the blueline, registering just 1 goal. On November 19, 2017, he was traded by Vityaz to Yugra, recapturing his form to play out the season with 6 points in 24 games.[4]
Having returned to the SHL with the Växjö Lakers for the 2018–19 season, Burström then returned to original hometown club, Skellefteå AIK, agreeing on a two-year contract on 11 April 2019.[5]
Following a lone season in the Deutsche Eishockey Liga (DEL) with the Schwenninger Wild Wings, Burström returned to the SHL after securing a three-year contract with IK Oskarshman on 16 April 2022.[6]