After posting 22 goals and 53 points in his rookie junior ice hockey season, he received the Jim Piggott Memorial Trophy for the top first-year WHL player.

The following season, Cozens, who had been playing on the wing as a rookie, moved back to his natural centre position to occupy a space abdicated by Sam Reinhart and Jack Eichel.

[4] Because there were limited hockey players in Whitehorse, Cozens often played against older opponents; when he was 12, he broke his leg after being boarded by an adult man in a house league game.

[10] He signed with the team that May,[11] but because he was below the WHL's age minimum for the upcoming 2016–17 season, he played for the Yale Hockey Academy in Abbotsford, British Columbia, instead.

Cozens added another two goals and four points in three CSSHL playoff games, and once the Yale season ended, he rejoined Lethbridge for the WHL postseason.

[18] In his first full season of junior ice hockey, Cozens posted 22 goals and 53 points in 57 regular-season games, and he received the Jim Piggott Memorial Trophy as the top rookie in the WHL.

[21] Cozens was a finalist for the CHL Rookie of the Year award at the end of the season, a title that ultimately went to Alexis Lafrenière of the Rimouski Océanic.

[26] Cozens finished his sophomore junior hockey season with 34 goals and 84 points in 68 regular-season games, and the NHL Central Scouting Bureau ranked him the No.

[34] Cozens told reporters that he wanted to "stay at an NHL level" during the Hurricanes' season, and by December 11, he led the team with 42 points (19 goals and 23 assists) in 28 games.

[36] By the time that concerns surrounding the COVID-19 pandemic forced the WHL to indefinitely postpone the 2019–20 season,[37] Cozens led Lethbridge with 38 goals and 85 points in 51 games.

[38] Cozens was named to the WHL Eastern Conference First All-Star Team at the end of the season,[39] and he was the runner-up for the Four Broncos Memorial Trophy, an award which went to Adam Beckman of the Spokane Chiefs.

Although Buffalo lost the game 6–4 to the Washington Capitals, Cozens recorded his first NHL point with an assist on Tobias Rieder's second-period goal.

[48] With Sam Reinhart's offseason departure and Eichel indefinitely sidelined due to a dispute with the Sabres' front office, an opportunity opened for Cozens to become a top-six centre for Buffalo during the 2021–22 season.

[56] He scored a controversial goal in the semifinals, tying the Canada-USA game with 0.00 left on the clock to take the match into overtime, where Canada won 6–5 to eliminate Team USA.

[62] He scored two goals and seven assists in seven World Junior games, winning a gold medal when Canada defeated Russia 4–3 in the championship match.

[64] He had eight goals and 16 assists, second among all participants, and was named to the tournament all-star team, while Canada took silver after a 2–0 shutout loss to the United States.

[66] Following the end of the 2023–24 NHL season, with the Sabres not qualifying for the 2024 Stanley Cup playoffs, Cozens accepted an invitation to join Team Canada at the 2024 IIHF World Championship.