Allen signed for Chelsea on his 17th birthday in September 1954 but never established his place in the Chelsea first-team. He scored 12 League goals for Chelsea in 44 appearances before joining Tottenham Hotspur in December 1959.
He was a key member of Tottenham's double winning side of 1960-61, forming a lethal goalscoring partnership with Bobby Smith. He scored 47 league goals in 119 appearances, but struggled to retain his place following the arrival of Jimmy Greaves.
He joined Queens Park Rangers in July 1965 and scored 55 goals in 123 starts and five substitute appearances. He was a member of the QPR team who became the first Third Division side to win the Football League Cup Final 1967.