He hails from Steinsland in Sund and started his career in local Skogsvåg IL. In 2010, he started playing regular senior football as he transferred to Hald FK. After half a season he went on to Løv-Ham,[2][3] which was soon merged to form FK Fyllingsdalen. He stayed with Fyllingsdalen until the summer of 2015, when he stepped up one tier to First Division club Åsane.
On 20 June 2021, Glesnes scored a late equalizer from well outside the 18-yard box to secure a 2–2 draw vs. Atlanta United.[11] In the first round of the 2021 MLS Cup Playoffs on 20 November, he scored a long-distance screamer in the 123rd minute versus the New York Red Bulls to seal a 1–0 victory and send the Union to the Eastern Conference semifinals.[12] On 8 December 2021, Glesnes signed a new, multi-year contract extension with the Union through the 2024 season.[13][14]
On 19 October 2022, Glesnes was awarded the MLS Defender of the Year Award, becoming the first Philadelphia Union defender to win the award.