Prior to being signed by the Maple Leafs as a free agent, Bozak had played two seasons at the University of Denver in the National Collegiate Athletic Association (NCAA).

In his final season, Bozak recorded 128 points (45 goals and 83 assists), and was awarded the Brett Hull Trophy as the BCHL's top scorer.

[4] On December 12, he tore the meniscus in his left knee in a game against the Minnesota State Mavericks, causing him to miss more than half of the Pioneers' season.

He had a strong training camp, though due to salary cap constraints, he was sent down to the Leafs' American Hockey League (AHL) affiliate, the Toronto Marlies.

On March 28, 2015, towards the conclusion of the 2014–15 season, Bozak scored his first career NHL hat-trick, as well as recording one assist to cap off a four-point performance, contributing to all four Toronto goals.

During the season, future superstar Connor McDavid, who was drafted in June 2015, stated that Bozak was his most comparable player due to the centre's good skating and "pass first" mentality.

[14][15] To the surprise of many, Bozak experienced career years following Kessel's departure, scoring at a pace much higher than predicted, and remaining on Toronto's top two lines.

[16][17][18] In an ironic turn of events, both Bozak and James van Riemsdyk (who also shared top line duties since joining the team) had more points and better analytics than Kessel at the Christmas break.

The following season, 2016–17, Bozak scored a career-high 55 points on a line with van Riemsdyk and rookie Mitch Marner, helping the team make the playoffs for the second time in his career.

After nine seasons with the Maple Leafs, Bozak left the club as a free agent, signing a three-year, $15 million contract with the St. Louis Blues on July 1, 2018.