Masur began playing tennis at the age of 8. In 1980, he won the boy's doubles title at the Australian Open. In 1981, aged 18, he won the Australian Junior Championships. He turned professional in 1982.

In 1983, Masur won his first top-level singles title at Hong Kong, and his first tour doubles title at Taipei. He also reached quarter-finals of that year's Australian Open, before being knocked-out by John McEnroe.

In 1987, Masur won his second career singles title at Adelaide and reached the Australian Open semi-finals, where he lost to eventual champion Stefan Edberg.

Masur won his third singles title in 1988 at Newport, Rhode Island.

In 1990, Masur helped Australia reach the final of the Davis Cup, compiling a 6-0 record in singles rubbers in the first round, quarter-finals and semi-finals. However he was left out of the team that played the United States in the final by captain Neale Fraser. The decision to leave Masur out of the final was fairly controversial at the time given the very significant role that he had played in getting Australia there, but was principally because the final was to be played on clay courts, which was not Masur's best surface. The US beat Australia 3-2 in the final.

1993 was possibly the best year of Masur's career. He reached the semi-finals of that year's US Open, where he lost to Cedric Pioline. He also reached his career-high rankings in both singles (World No. 21) and doubles (No. 8) that year. He captured doubles titles in Milan and Stuttgart that year, which proved to be the final top-level titles of his career.

Masur retired from the professional tour in 1995, having won 3 singles titles and 16 doubles titles. His career prize-money totalled US$3,134,718.

Since retiring from the tour, Masur has served as coach of Australia's Davis Cup team.