Playing the position of defenceman, Norwood was drafted in 1979 by the Quebec Nordiques. He also played for the Washington Capitals, St. Louis Blues, Detroit Red Wings, New Jersey Devils, Hartford Whalers, and Calgary Flames. After his stint with the Flames, he retired from professional play in 1994, only to return to active play a year later in the International Hockey League for two seasons before hanging up the skates for good.