Greenway has already made an impact early in his hockey career. The second-round pick (No. 50) of the Minnesota Wild in the 2015 NHL Draft won a gold medal for the United States at the 2014 World U-17 Hockey Challenge and 2015 IIHF World Under-18 Championships. He excelled at Boston University and played at the 2017 IIHF World Championship after his sophomore season. The 6-foot-6 forward became the first African-American player to compete for the U.S. in the Olympics at the 2018 PyeongChang Games.

Born in Canton, New York, Greenway played high school hockey at Shattuck-St. Mary's in Minnesota before joining the USA Hockey National Team Development Program for two seasons (2013-15). As a junior at BU, he was selected to the Hockey East Third All-Star Team after earning 30 points for the second consecutive season.

Greenway signed a three-year contact with the Wild on March 26, 2018, and made his NHL debut the next day in a 2-1 loss to the Nashville Predators. He had one assist in five games with the Wild, then scored one goal in Minnesota's five-game loss to the Winnipeg Jets in the Western Conference First Round, becoming the first player to compete in the Olympics, the NCAA Tournament and the Stanley Cup Playoffs in the same season.

Greenway signed a three-year, $9 million contract with the Wild on Jan. 31, 2022. He had seven points (two goals, five assists) in 45 games of the 2022-23 season before he was traded to the Buffalo Sabres on March 3, 2023, for a second-round pick in the 2023 NHL Draft and a fifth-round pick in the 2024 NHL Draft.

He signed a two-year, $8 million contract with the Sabres on March 5, 2025.