The Calgary Flames selected Kulak, a defenseman born in Stony Plain, Alberta, in the fourth round (No. 105) of the 2012 NHL Draft after his first full season with Vancouver of the Western Hockey League. He spent two more seasons with Vancouver and had 60 points (14 goals, 46 assists) in 2013-14 before signing a three-year, entry-level contract with the Flames on March 18, 2014.

Kulak started his first pro season in the ECHL, was promoted to Adirondack of the American Hockey League and made his NHL debut on April 11, 2015, in Calgary's season finale. Kulak spent most of 2015-16 in the AHL, playing eight games with the Flames, and earned 21 games with Calgary in 2016-17, earning his first NHL point with an assist on Oct. 18, 2016.

He became a regular on the Flames' third defensive pair in 2017-18. He had eight points (two goals, six assists) in 71 games. He was traded to the Montreal Canadiens on Oct. 1, 2018. In his first season in Montreal, Kulak had 17 points (six goals, 11 assists) in 57 games.

Kulak had 25 points (seven goals, 18 assists) for the Edmonton Oilers in 2024-25, his third consecutive season playing all 82 games with 135 shots on goal and playing his 500th NHL game Oct. 12 against the Chicago Blackhawks. He was acquired by the Pittsburgh Penguins with goalie Stuart Skinner in the trade that sent Tristan Jarry to the Oilers on Dec. 12, 2025. The Penguins then traded Kulak to the Colorado Avalanche on Feb. 24, 2026, for defenseman Samuel Girard and a second-round pick in the 2028 NHL Draft.

Kulak scored at 3:52 of overtime to send the Avalanche to the Western Conference Final with a 4-3 win against the Minnesota Wild in Game 5 of the Western Conference Second Round.