Moore played with the Montreal Canadiens from 1951 to 1963. He started playing with the Montreal Jr. Royals for three seasons from 1947 to 1950, and made his debut with the Montreal Canadiens in the middle of the 1951–52 season. Moore had played on two Memorial Cup winners, one with the Montreal Royals in 1949 and Montreal Junior Canadiens the following year. He was known for his hard accurate shot and his ability to stickhandle the puck. He twice won the Art Ross Memorial Trophy as the league's leading scorer. Moore broke Gordie Howe's record of 95 total points in a regular season play with 41 goals and 55 assists.