Eastham began his career with Northern Irish side Ards before moving back to England with Newcastle United in 1956. He became a fine Inside forward for the "Magpies" but then shocked the club by demanding a move away which Newcastle rejected. Eastham took his case to the courts and won his case moving to Arsenal. Eastham spent six seasons at Highbury making 223 appearances scoring 41 goals for the "Gunners" before joining Stoke City in 1966. His experience helped Stoke enjoy a successful spell in the early 1970s and Eastham scored the winning goal in the 1972 League Cup Final. He also spent time coaching in South Africa playing for Hellenic before returning to Stoke to become assistant manager to Tony Waddington. When Waddington resigned in March 1977 Eastham was appointed manager but was unable to prevent Stoke being relegated in 1976–77 and after failing to mount a promotion challenge the following season Eastham was sacked in January 1978. He then returned to coach in South Africa.