A wing half, Goodwin was signed as a trainee from Cheshire Schoolboys by Manchester United on 1 October 1953 as one of the Busby Babes. He made his senior debut for the club on 20 November 1954 against Arsenal. He helped the club win the 1956 and 1957 league championships, and was a member of the United team that made a comeback from the Munich air disaster that claimed the lives of eight players and ended the careers of two others, to reach the 1958 FA Cup Final, losing 2–0 to Bolton. He was not on the plane to Munich, having not been selected in the squad for the quarter-final second leg tie with Red Star Belgrade of Yugoslavia.
In his United career, he scored eight goals in 107 appearances. He was signed by Leeds United on 16 March 1960 for £10,000. In the 1963–64 season, a collision with former Leeds team-mate John Charles in an FA Cup tie against Cardiff City caused him to suffer a triple fracture of his leg, eventually resulting in his retirement from playing on 1 December 1964. He scored two goals in 120 appearances for Leeds.
Goodwin went on to become a player-manager at Scunthorpe United, although he did not play many games due to his injury, making six appearances and scoring one goal for the club. He left Scunthorpe United on 1 June 1966 and then became manager for the New York Generals (appearing in one match) and Brighton & Hove Albion, before becoming manager of First Division club Birmingham City. It was at Birmingham where he introduced the young Trevor Francis into league football. At Birmingham, he was also known for introducing yoga, psychological testing and other new training techniques.
From 1976 to 1978, he was the first coach and president of the Minnesota Kicks of the North American Soccer League. He remained as president of the club until June 1981. He coached the team for portions of the 1980 and 1981 seasons. Goodwin and wife Sylvia retired to the Pacific Northwest in the early 1990s.