Kulak was then recalled to Calgary and made his NHL debut on April 11, 2015, in the Flames' final regular season game, a 5–1 loss to the Winnipeg Jets.

[5] On October 18, 2016, against the Buffalo Sabres, Kulak recorded his first career point, assisting on Michael Frolík's first period goal.

On August 28, 2017, the Flames re-signed Kulak (who was a restricted free agent) to a one-year, two-way contract worth $650,000.

[7] On October 1, 2018, the Flames traded Kulak to the Montreal Canadiens in exchange for Matt Taormina and Rinat Valiev.

[12] Over the following seasons, Kulak typically played on the team's third pairing, notably participating in the Canadiens' deep run to the 2021 Stanley Cup Finals.

In the final season of his contract, Kulak assumed an increasingly important role in the team's defence group, as a result of departures and injuries to other key players.

[13] On March 19, 2022, he played what would prove to be his final game as a Canadien, scoring a goal in a 5–1 rout of the Ottawa Senators.