Dobson began his career at Sheffield Wednesday, who finished the 1961–62 season in sixth place in the First Division under the stewardship of Vic Buckingham. He scored one goal in the 1961–62 Inter-Cities Fairs Cup, in a 5–2 home victory over Lyon; Wednesday went on to reach the quarter-finals, where they were knocked out by Barcelona after a 2–0 defeat at Camp Nou. The "Owls" again posted sixth-place finishes in 1962–63 and 1963–64, before slipping to eighth position in 1964–65 under new boss Alan Brown. They dropped down to 17th spot in 1965–66. In league and cup competitions, Dobson scored 52 goals in 193 first team appearances at Hillsborough. He moved on to Tom Johnston's Huddersfield Town for a £20,000 fee in August 1966. He made a big impact with 17 goals in 39 games in the 1966–67 season; strike partner Tony Leighton also claimed 20 goals to help Town to a sixth-place finish in the Second Division. Dobson finished as the club's top-scorer with 14 goals in 47 games in the 1967–68 campaign, including one in the League Cup semi-final defeat to Arsenal at Highbury. He then was top-scorer for a second time as he bagged 11 goals in 43 matches in the 1968–69 season, as new boss Ian Greaves took the "Terriers" to within 10 points of promotion. Dosbon built a successful partnership with Frank Worthington and claimed nine goals in his 33 appearances as promotion was achieved with a first-place finish in 1969–70. However, he appeared just 12 times in the 1970–71 campaign, claiming one goal, and departed Leeds Road at the end of the season. He also played four Third Division games on loan at Pat Saward's Brighton & Hove Albion in January 1972. Dobson signed with Don Megson's Bristol Rovers as a player-coach. The "Pirates" finished fifth in the Third Division in 1972–73, before winning promotion in second place in 1973–74. The "Gas" avoided relegation by a three-point margin in 1974–75 and then by a five-point margin in 1975–76. He scored four goals in 63 league and cup games during his time at the Eastville Stadium.
Dobson left Bristol Rovers to become the youth coach at Coventry City. He acted as Port Vale's coach on a voluntary basis from December 1983 to March 1984 to help out former teammate turned manager John Rudge, before managing West Riffa in Bahrain. Later he became the coach of Qatar side Al Rayyan, before returning to England to take up the youth coach position at Aston Villa. He also was the youth coach at Portuguese Sporting Lisbon, the youth coach and chief scout at Gillingham, a coach back at the Coventry City youth set-up, before taking up the position as first team coach of Kuwaitee Al Arabi. In September 1995 he returned to Port Vale as a temporary coach, and departed Vale Park again in May 1996. He went on to coach Oman at the 1997 FIFA U-17 World Championship in Egypt. He later worked as chief scout at Stoke City, and was credited with the discovery of England international goalkeeper Ben Foster. He was appointed chief scout at the Britannia Stadium in 2000 by director of football John Rudge, but left to scout for Watford in June 2005, only to return to the "Potters" by 2008. He left the club in June 2013.
