He broke into the NHL in the 1964–65 season, playing 23 games with the Boston Bruins as the backup to Eddie Johnston. He lost his spot the following season to Gerry Cheevers. Norris was subsequently dealt to the Chicago Black Hawks in the Phil Esposito trade, where he played ten games over two seasons. Norris spent significant time in the minor leagues before eventually joining the Los Angeles Kings as the backup to Denis DeJordy. After spending the 1971–72 season in the minor leagues, Norris joined the Alberta Oilers of the WHA.