Cooper reached final in 1954 Junior Wimbledon losing to Ramanathan Krishnan in Boys' Singles tournaments.

Cooper won his first Grand Slam singles title at the 1957 Australian Championships where he defeated compatriot Neale Fraser in the final in four sets.

[6] Cooper was ranked amateur world number one in 1957 by Ned Potter,[7] Lance Tingay,[8] Adrian Quist[9] and Yvon Petra.

He successfully defended his Australian singles title after a straight-sets victory in the final against Malcolm Anderson.

[15] Cooper won his first professional match against Pancho Gonzales at Perth on grass in the Ampol world series of tournaments, and defeated Sedgman in the semifinal, but lost the final to Hoad.

In the fall of 1959, he won the Slazenger Professional tournament at Eastbourne, England on grass, defeating Trabert in the semifinal and Hoad in the final in best-of-five set matches.

[21] In the Queen's Birthday Honours List of 2007, he was appointed an Officer of the Order of Australia (AO) for his service to tennis.

An estimated crowd of five thousand unruly people surrounded St. Paul's Presbyterian Church in Brisbane to try to catch a glimpse of the couple.