At 17 years and 256 days old, Allen's appearance for West Ham United against Arsenal in the 1980 FA Cup Final made him the youngest player to appear in an FA Cup final at Wembley Stadium, though James Prinsep played at a younger age at Kennington Oval in 1879. Both records have since been broken. His debut for West Ham had come on 29 September 1979 when 32 days after his 17th birthday he had appeared for the club in their 2–1 home win over Burnley in the Second Division. He collected a Second Division title medal for the Hammers in 1980–81 and helped re-establish them as a First Division side. He made 152 league appearances for the Hammers, scoring six goals, before a £400,000 fee took him to their London rivals Tottenham Hotspur on 19 June 1985.