Stapleton was a subtle, intelligent player, held in esteem by many for his strategic thinking and accurate passing on the pitch. He started his career with Arsenal, joining them in 1972 as an apprentice. He made his first-team debut in 1975 against Stoke City, and would go on to form a potent striking partnership with Malcolm Macdonald; the two scored 46 goals between them in 1976-77. He was Arsenal's top scorer for the three following seasons, and helped the Gunners reach a trio of FA Cup finals; Stapleton scored one of the goals in Arsenal's 1979 3-2 win over Manchester United, and scored 108 goals in 300 appearances in total for the Gunners.

Stapleton went on to move to Man United in 1981 for £900,000 (a fee set by tribunal after the two clubs could not agree). He would help United win the 1983 and 1985 FA Cups. He left United in 1987, after scoring 78 goals for the club in 365 matches.

He went on to play for Ajax Amsterdam, before returning to England with Derby County and Blackburn Rovers. He also won 71 caps for the Republic of Ireland, scoring a then record 20 goals. In 1991 he became player-manager of Bradford City, where he spent three seasons, and later went to the United States to coach Major League Soccer side New England Revolution.

In the 2003-04 season he briefly returned to English football as a specialist coach of Bolton Wanderers. The Bolton manager Sam Allardyce wanted Stapleton to enhance the skills of the strikers at the club and saw the Irishman as an ideal candidate, given his successful playing career.