A right-hander, Anderson started playing tennis when he was 8 and became serious about the sport at 16. His two best seasons statistics-wise were 1957 and 1958 when as an amateur he twice achieved a ranking of World No. 2.
In 1958, Anderson was a finalist at both the Australian Championships and US Championships, losing both times to Cooper. Anderson turned professional in late 1958 and went on to win the Wembley Championship in 1959, with a thrilling 5-set victory over former 3-time US Pro champion, Pancho Segura. Anderson did not appear in another major final until 1972, when at age 36, he was a finalist at the Australian Open, losing to Ken Rosewall. In 1973, he captured the Australian Open doubles title along with John Newcombe.