After 74- and 80-point seasons with Youngstown of the United States Hockey League from 2013-15, Connor was selected by the Winnipeg Jets in the first round (No. 17) of the 2015 NHL Draft.

Before making his NHL debut, Connor played one season at the University of Michigan, where he was named the Big Ten Player of the Year as a freshman after finishing with 71 points (35 goals, 36 assists) in 38 games and finished second to Jimmy Vesey (Harvard) for the Hobey Baker Award, given to the top player in NCAA hockey.

Connor signed a three-year, entry-level contract with the Jets on April 11, 2016. He played most of the 2016-17 season for Manitoba of the American Hockey League, but played 20 games with the Jets and scored his first NHL goal in a 4-1 win against the Dallas Stars on Oct. 27, 2016.

Connor began the 2017-18 season with Manitoba, but was recalled by the Jets on Oct. 16, 2017, and scored a goal the next night against the Columbus Blue Jackets; he went on to lead NHL rookies with 31 goals, helping the Jets to their best season in franchise history (52-20-10, 114 points). He also had 10 points (three goals, seven assists) in 17 games during the Stanley Cup Playoffs for the Jets, who advanced to the Western Conference Final before losing to the Vegas Golden Knights in five games.

He showed his rookie season was no fluke by scoring 34 goals in 2018-19 and bettering that total in his third season.

Connor in 2021-22 became the first Jets/Atlanta Thrashers player to win the Lady Byng Trophy, awarded annually to the NHL player voted to best combine sportsmanship, gentlemanly conduct and ability. He led Winnipeg and set NHL career highs in goals (47), assists (46) and points (93) in 79 games this season, and was assessed two minor penalties, the fewest among the top 100 scorers in the NHL (minimum, 50 games), while leading Jets forwards in ice time per game (21:47).

Connor tied an NHL record by scoring in his sixth straight season opener, a 5-3 loss to the Calgary Flames on Oct. 11, 2023. He joined Cam Atkinson of the Columbus Blue Jackets (2014-15 to 2019-20), Yvan Cournoyer of the Montreal Canadiens (1973-74 to 1978-79) and Mud Bruneteau of the Detroit Red Wings (1940-41 to 1945-46).

Playing against his hometown team, the native of Shelby Township, Michigan, moved to 502 points (252 goals, 250 assists) in 541 NHL games, getting No. 500 in the NHL during Winnipeg's 6-2 win against the Detroit Red Wings at Little Caesars Arena on Oct. 30, 2024. He became a seven-time 30-goal scorer in Winnipeg's 6-2 win against the Boston Bruins at TD Garden on Jan. 30, 2025, tying Ilya Kovalchuk for the most in Jets/Atlanta Thrashers history.

Connor had one assist in three games for the United States at the 4 Nations Face-Off and was named to Team USA's roster for the Olympic Winter Games Milano Cortina 2026. He signed an eight-year, $96 million contract with the Jets (average annual value of $12 million) on Oct. 8, 2025, after leading Winnipeg and setting NHL career-highs in points (97) and assists (56). He also led the Jets with 17 points (five goals, 12 assists) in 13 playoff games.