Brännström earned his first NHL recall on 5 May 2018 to help the Golden Knights during their 2018 Stanley Cup playoffs run, but did not appear in any games.

[6] Brännström attended the Golden Knights 2018 training camp but was reassigned to the American Hockey League (AHL) on 27 September 2018.

[8] On 25 February 2019, Brännström was traded to the Ottawa Senators as part of a package to acquire forwards Mark Stone and Tobias Lindberg.

On 21 November 2020, in his eighth game with the Tigers, Brännström was punched in the face by EHC Biel's Damien Brunner during a line change in the third period.

[18] On 11 December 2020, Brännström left the Tigers after having appeared in 10 games (8 points) to return to Ottawa for the start of training camp.

[30] During the 2022–23 season Brännström's play improved as he was often called on to replace injured defenceman after a number of them, including regulars Thomas Chabot and Jakob Chychrun, were affected.

A restricted free agent at the end of the season, Brännström signed a one-year contract extension with the Senators on 1 July 2023.

[33] Ending his five-year tenure with the Senators, Brännström was soon signed as a free agent to a one-year, $900,000 contract with the Colorado Avalanche on 2 July 2024.

[34] He attended the Avalanche's 2024 training camp and on the last day was traded to the Vancouver Canucks for defenceman Tucker Poolman and a 2025 fourth-round draft pick.

Brännström was traded to the New York Rangers, along with J. T. Miller, in exchange for Filip Chytil, Victor Mancini, and a 2025 first-round pick.