[5] Undrafted in the NHL entry draft, Gleason signed a three-year entry-level contract with the Dallas Stars on September 13, 2018.

[10] As a free agent from the Stars after five seasons within the organization, Gleason was signed to a two-year, $1.55 million contract with the Edmonton Oilers on July 2, 2023.

[11] Gleason impressed Oilers staff during the 2023–24 preseason and was considered a dark horse to make the final roster.

[12] In spite of this, Gleason was a late cut from their roster and was re-assigned to their AHL affiliate, the Bakersfield Condors, prior to the start of the season.

His other cousin, Jacob, played with the Metro Jets in the United States Premier Hockey League (USPHL).