Compher, a forward chosen by the Buffalo Sabres in the second round (No. 35) of the 2013 NHL Draft, enjoyed a decorated career at the University of Michigan.

During his first season (2013-14), Compher led the Wolverines and all Big Ten Conference freshmen with 31 points (11 goals, 20 assists). Following his sophomore year, Compher was elected captain for the 2015-16 season. At the 2015 NHL Draft, the Sabres traded Compher, forwards Mikhail Grigorenko and Nikita Zadorov, and a second-round pick to the Colorado Avalanche for forwards Ryan O'Reilly and Jamie McGinn.

Compher continued to excel at Michigan in 2015-16, leading the nation with 47 assists, the most by a Wolverines player since Brendan Morrison's school record of 57 in 1996-97, and finishing second in Division I with 63 points. He signed a three-year contract with the Avalanche on April 25, 2016, and began his professional career by putting up 30 points (13 goals, 17 assists) in 41 games with San Antonio of the American Hockey League before his recall to Colorado on March 1, 2017. In his eighth game with the Avalanche, Compher earned his first NHL point, the game-winning goal in a 3-1 victory against the Detroit Red Wings on March 15.

In 2017-18, his first full NHL season, Compher had 23 points (13 goals, 10 assists) in 69 games to help Colorado clinch the second wild card into the Stanley Cup Playoffs from the Western Conference one season after the Avalanche finished last in the NHL (22-56-4). In the 2019 playoffs, he finished tied for third on Colorado with four goals in 12 games (Colin Wilson, Matt Nieto).

Compher signed a five-year, $25.5 million contract with the Detroit Red Wings on July 1, 2023, an average annual value of $5.1 million, coming off an NHL career-high 52 points (17 goals, 35 assists) in 82 regular-season games in 2022-23, and two points (one goal, one assist) in seven playoff games. He also won the Stanley Cup with the Avalanche in 2022.