Although born in Aberdeen, Aberdeenshire, Dawson started his career as a trainee with Manchester United. He helped the club win the league title in 1957. He left United for Preston North End in 1961 after scoring 54 goals in all competitions for the Red Devils. At Preston, he became known as the "Black Prince of Deepdale" and featured in their 1964 FA Cup Final team. He scored in the final as Preston lost 3–2 to West Ham United. He later played for Bury, Brighton and Brentford.
Dawson was the last player to score a hat trick in an FA Cup semi-final, back in 1958 when Manchester United beat Fulham 5–3 in a replay at Highbury.