Upon his graduation from East High School in Rochester, New York, in 1923, Harrison began playing, coaching, and organizing semi-professional basketball, working for the Rochester Seagrams and later the Rochester Ebers.
In 1945, with his brother Jack Harrison founded his own semi-pro team, the Rochester Pros; in 1946, the franchise changed its name to Rochester Royals and began play in the National Basketball League (NBL). Harrison coached the NBL iteration of his team for three years, leading the Royals to three straight NBL finals (of which the team won one, over the Sheboygan Redskins in 1946) and amassing a record of 99 wins and 43 losses. In 1946, Harrison, as team owner, signed Long Island University standout Dolly King; King became the first African-American to have played in the league since 1943 and the first to see substantial playing time for his team.
After the 1947-48 season, Harrison moved his team to the Basketball Association of America (BAA), and, after his team spent one year in the BAA, served on the committee that brokered the merger of the BAA and NBL and produced the National Basketball Association, the extant North American major professional basketball league.
Harrison coached the Royals through the 1954-55 season and retired having led his team to five NBA divisional titles and the 1951 NBA championship. He remained owner of the Royals for three more years, relocating the team to Cincinnati, Ohio (the team currently exists in Sacramento, California, as the Kings), prior to the 1957 season and selling in 1958.
In view of his having been a member of the boards of directors of the NBL, BAA, and NBA, having helped broker the merger of the NBL and BAA, and having been a proponent of the introduction of the 24-second shot clock, Harrison was inducted into the Basketball Hall of Fame as a contributor in 1980; in 1990 he was inducted into the International Jewish Sports Hall of Fame.