Azam was introduced to squash by his older brother, Hashim Khan, who was also one of the world's top squash players in his time. After winning the British Open in 1951, Hashim recruited Azam as his practice partner in the summer of 1952. (The British Open was considered to be the effective world championship of the sport at the time.) Azam progressed so quickly under his older brother's tutelage that he would go on to face Hashim in the final of the 1954 British Open, which Hashim won in five sets. Azam was also runner-up to Hashim at the British Open in 1955 and 1958. Azam then went on to win four consecutive British Open titles in 1959-62. He also won the US Open in 1962. Later that year, a ruptured Achilles tendon effectively ended his career.
After his playing career, Azam settled in the United Kingdom. Today, he owns and runs the New Grampians Squash Club in London. His granddaughter Carla Khan is currently a professional squash player on the international circuit.