Junior The
Portland Winterhawks of the
Western Hockey League (WHL) selected Kozak in the sixth round, with the 121st overall pick, of the 2017 WHL bantam draft, and he signed with the team the following August. Kozak played in three
junior ice hockey games for Portland during the
2018–19 season before he was reassigned to the Cougars. He made two more appearances with the team that season: once in December as a replacement for
Cody Glass, who was participating in the
2019 World Junior Ice Hockey Championships; and again in March as a
black ace for the WHL playoffs. He was held pointless in all five regular season and four postseason games. Kozak joined the Winterhawks in full for the
2019–20 season. He recorded his first junior point on September 21, assisting on Cross Hanas's goal against the
Tri-City Americans. Kozak scored his first two goals for the Winterhawks the following week, during Portland's 4–2 victory over the
Vancouver Giants on September 27. One of the younger players on the team that season, Kozak credited the Winterhawks' coaching staff for improving his confidence, and through the first 29 games of the season, Kozak posted four goals and 10 points. Playing both
centre and
wing, Kozak finished his rookie season with 13 goals and 29 points in 63 games, being named the Winterhawks Rookie of the Year. Ahead of the
2020–21 WHL season, the
NHL Central Scouting Bureau named Kozak a C-grade prospect on their preliminary players-to-watch list. The
COVID-19 pandemic and its effects on ice hockey limited the Winterhawks to 24 games that year, six of which Kozak missed due to a hip flexor injury early in the season. After failing to record a point through his first five games, he steadily improved throughout the remainder of the shortened season, finishing with three goals and 11 points in 18 games. He also played in one game with the
Dauphin Kings of the
Manitoba Junior Hockey League while waiting for the WHL season to begin. At the end of the season, the
Buffalo Sabres of the
National Hockey League (NHL) selected Kozak in the seventh round, with the 193rd overall pick, of the
2021 NHL entry draft. After attending training camp with the Sabres, Kozak returned to Portland for the
2021–22 season. There, he was named co-
captain of the Winterhawks, alongside Clay Hanus. Centering Portland's top line, Kozak overcame a slow start. After recording only one assist in his first eight games, he added 20 goals and 39 points through his next 30 games, setting career highs in goals and points halfway through the season. He attributed his performance to greater confidence, saying that he was "not as worried about making mistakes" as he had been in prior seasons. Kozak finished the season fourth in team scoring, with 32 goals and 69 points in 66 games. At the end of the year, he was named to the WHL U.S. Division Second All-Star Team.
Professional The Sabres signed Kozak to a three-year,
entry-level contract on August 19, 2022. After training camp, he was assigned to the
Rochester Americans, Buffalo's
American Hockey League (AHL) affiliate, to begin his professional hockey career. Kozak scored his first professional goal on October 21, against
Syracuse Crunch goaltender
Maxime Lagacé. Despite missing time with injury, Kozak finished the
2022–23 season with five goals and 10 points in 55 regular season games. Despite his middling points, Kozak received praise from the Americans coaching staff for his physicality and his abilities on the
penalty kill. Kozak added another two goals and five points in 14 postseason games during the
2023 Calder Cup playoffs. Kozak opened the
2023–24 season on the Americans. He spent the entire season in the AHL, struggling to find his rhythm as he battled through injury. These recurring lower-body injuries limited Kozak to only 41 games, during which he posted five goals and 12 points. Kozak was injured again at the Prospects Challenge rookie tournament, causing him to miss all of training camp ahead of the
2024–25 season. When he returned, he found success in Rochester, matching his career high with five goals through his first eight games of the AHL season. After recording six points through 14 games, Kozak was promoted to the Sabres on November 29, 2024, as a replacement for an injured
Sam Lafferty. He made his NHL debut on December 5, playing on the fourth line with
Beck Malenstyn and
Nicolas Aubé-Kubel against the
Winnipeg Jets. Kozak appeared to score against
Connor Hellebuyck in his debut, but the goal was called back for
goaltender interference. Kozak scored his official first goal two days later against the
Utah Hockey Club. The Sabres signed Kozak to a three-year, $2.325 million extension on July 1, 2025. ==Career statistics==