Cossa was drafted in the 2017 WHL bantam draft in the second round, thirty-sixth overall, by the Edmonton Oil Kings. As a rookie during the 2019–20 season, he posted a 21–6–3 record, with a .921 save percentage in 33 starts to finish tied for third-best in the WHL before the season was cancelled due to the COVID-19 pandemic. In December 2019, he posted a 6–1–1–0 record, with a 1.98 goals against average (GAA), and stopped 222 of 238 shots for a 933 save percentage, and was named the WHL Goaltender of the Month.[1]
During the 2020–21 season, he posted a 17–1–1 record. He led the WHL in goals against average (1.57), save percentage (.941) and shutouts (4). By winning his first 12 starts of the year, he set a franchise record for consecutive wins in a single season by a goaltender.[2] In April 2021, he posted a 6–0–0–1 record, with a 1.61 GAA, .936 save percentage and two shutouts, and was named WHL Goaltender of the Month.[3] He was a finalist for the WHL Goaltender of the Year award, and led the Oil Kings to the best record in the league.[4]
On July 23, 2021, Cossa was drafted 15th overall by the Detroit Red Wings in the 2021 NHL entry draft.[5] On August 14, the Red Wings signed Cossa to a three-year, entry-level contract.[6] Cossa was named the WHL Goaltender of the Month for the month of October 2021. He posted a 6–2–1–1 record in 10 games, with a 1.58 GAA, a league-best .943 save percentage, and one shutout.[7]
Cossa made his professional debut for the Grand Rapids Griffins of the AHL on October 19, 2022. He made 21 saves and earned the win in a 3–2 victory over the Milwaukee Admirals. He was reassigned to the Toledo Walleye of the ECHL the next day.[8] On January 21, 2023, Cossa earned his first professional shutout in a 5–0 victory against the Iowa Heartlanders.[9] Along with John Lethemon, Cossa helped lead the Walleye all the way to conference semifinals where they eventually lost to the league's top-seeded Idaho Steelheads 4 games to 1.[10]
During the 2023–24 season, in his first full season in the AHL, he posted a 22–9–9 record with a 2.41 GAA, .913 save percentage and two shutouts in 40 regular season games. During the 2024 Calder Cup playoffs he appeared in nine playoff games, and posted a 5–4 record, with a 2.72 GAA and .900 save percentage, as he helped the Griffins reach the Central Division Finals.[11][12]
On December 2, 2024, Cossa was recalled by the Red Wings under emergency conditions, after placing Alex Lyon on the injured reserve retroactive to November 27, and Cam Talbot suffering a lower-body injury on December 1, in a game against the Vancouver Canucks. He began the 2024–25 season with the Griffins, where he appeared in 14 games and posted a 9–4–1 record, with a 2.21 GAA and .925 save percentage.[13][14] Cossa subsequently made his NHL debut on December 9, entering in relief of Ville Husso after the first period.[15] He stopped 12 of 14 shots faced en route to a 6–5 shootout victory over the Buffalo Sabres, becoming the 11th goalie in Red Wings history to earn a win in their NHL debut.[16]