Bennett is a native of Holland Landing, Ontario. He played youth hockey for the York Simcoe Express and then the Toronto Marlboros. In 2012, he was selected ninth overall by the Kingston Frontenacs in the Ontario Hockey League (OHL) Priority Selection Draft. He appeared in 40 games with the Frontenacs in 2012–13 and recorded 40 points to earn a place on the OHL's Second All-Rookie Team. Bennett earned a place with the Canadian under-18 national team for the 2013 World U18 Championship. He recorded three goals and four assists in seven games as Canada won the gold medal with a 3–2 victory over the United States in the final. He also played on the gold medal-winning Canadian team at the 2013 Ivan Hlinka Memorial Tournament.
Returning to the Frontenacs for the 2013–14 season, Bennett was among the League's scoring Leaders by mid-season. He stood fourth with 66 points in 40 games and was on a 24-game scoring streak when he suffered a minor injury that prevented him from participating in the 2014 CHL Top Prospects Game. In its mid-season ranking, the NHL Central Scouting Bureau named Bennett as the top North American prospect for the 2014 NHL Entry Draft. One of the Bureau's scouts praised Bennett's offensive ability: "His puckhandling and playmaking are excellent and he has one of the best shots in this year's draft class. He has scored several goals from the high slot and coming in off the wing and has been very effective on the power play." He retained the top spot in NHL Central Scouting's final ranking after recording nine points in seven post-season games for Kingston despite playing through a groin injury. He finished the regular season with 36 goals and 91 points in 57 games played. He was named to the OHL's Third All-Star Team.
The Calgary Flames selected Bennett with the fourth overall selection at the 2014 NHL Entry Draft, and he quickly signed a three-year, entry-level contract with the club. He attended the team's training camp prior to the 2014–15 season, where he injured his shoulder. Bennett admitted that the shoulder had ailed him the previous season in Kingston, but he initially failed to disclose the issue, stating he "wanted to play" and would "do anything to make the team and play in the NHL this year." Instead, Bennett's injury required surgery to repair, which the Flames said would prevent him from playing for several months. After missing five months of action, Bennett was medically cleared to play and was reassigned to Kingston on February 21, 2015. He scored 24 points in 11 regular season games for Kingston and added 3 assists in the playoffs as the Frontenacs were quickly eliminated in the OHL playoffs.
The Flames recalled Bennett to Calgary after Kingston's season ended, and he made his NHL debut in the team's final regular season game on April 12, against the Winnipeg Jets. He recorded his first career point on his first shift – 33 seconds into the contest – assisting on a Michael Ferland goal in a 5–1 loss. Bennett was a regular player for the Flames in the 2015 Stanley Cup playoffs, and recorded his first NHL goal on April 19, a game-winner against Eddie Läck of the Vancouver Canucks in a 4–2 victory. Bennett played all 11 post-season games for the Flames in which he scored 3 goals and added an assist. On January 13, 2016, Bennett scored 4 goals against the Florida Panthers. At 19 years and six months, he became the youngest player in Flames history to score a hat-trick, and the third-youngest player in NHL history to score 4 goals in a game. Bennett scored his first three goals within the first 17 minutes of the game, and added a fourth goal in the last minute of the 3rd period.
On September 6, 2017, the Flames re-signed Bennett to a two-year, $3.9 million contract worth $1.95 million annually.
