Radford came up through the Arsenal youth system, joining them in 1962 at the age of 15. Two years later, he signed a professional contract, then made his first-team debut that March. It was his only appearance of the 1963-64 season, but he featured more the following season, leading up to his record-breaking performance against Wolves at the age of 17 years and 315 days.
He was a key component of the club's Double-winning 1970-71 season, scoring 21 league goals as Arsenal claimed the title and providing the assists for both goals in their 2-1 win over Liverpool in that season's FA Cup final. He also helped them win the 1969-70 Inter-Cities Fairs Cup.
Radford remained with Arsenal until 1976, having scored 149 goals--enough to take second place on the club's all-time scorer list at the time, behind Cliff Bastin (178), though they have since been overtaken by Ian Wright (185) and Thierry Henry (226).