Graham turned pro as a teenager and had much success on the Australasian circuits in his youth, winning several tournaments.

At young age he used to pass the Wattle Park nine-hole golf course when cycling to school and one day the local professional John Green offered him a job in the shop during week-ends.

[8] In February 1969, Graham's finished in a tie for third at the two-round $3,500 Amoco Open at Forbes, New South Wales, four back of champion Tony Mangan.

[11] The following week at the Singapore Open he was one back of leaders Tomio Kamata of Japan and Guy Wolstenholme of England after three rounds.

Graham made up one stroke on the leaders during the back nine to enter a playoff with them, but lost to Kamata on the third extra hole.

[6][15] At that point Graham said his goal was to qualify for the Alcan Open, a limited-field tournament on the PGA Tour to be held in September in the United States.

He shot a second round 69 to move into a tie for fifth, behind Guy Wolstenholme, Bruce Devlin, Peter Thomson, and Gary Player.

[26] In December he played the Caltex Tournament at Paraparaumu Links Golf Course in Wellington, New Zealand, and finished in a tie for fourteenth.

Graham birdied the final two holes to win by three over New Zealander Walter Godfrey and four over pre-tournament favorite Tommy Aaron.

Late in November 1970 it was announced that Graham would represent Australia at the 1970 World Cup with Bruce Devlin.

Crampton, as well as Kel Nagle and Peter Thomson, had declined to play[3] and The Australian PGA threatened not to send a team if Graham was not included.

It took a 54-hour plane trip for both Graham and Devin to reach Argentina and both were "tired" once they started playing the first round.

In the individual competition, Graham was in solo second, one behind leader, home legend Roberto De Vicenzo, while Devlin was tied for third.

[43] At the beginning of the final round both Graham and his partner Devlin played poorly shooting 35 and 36, respectively, over the course of the par-35 front nine.

[43] According to the Papua New Guinea Post-Courier, however, at the end of the front nine the Australians "were assured of victory over their nearest rivals Argentina."

At 544, they beat the team record set by Arnold Palmer and Jack Nicklaus at the 1966 Canada Cup by four shots.

[45] Another controversy with Graham involved was reported during the inaugural 1985 Dunhill Cup at the Old Course at St Andrews, Scotland.

At the time, Marsh had recently been made an MBE for services to golf and was for six years chairman of the PGA Tour of Australasia.

[47] In 1976, he won twice on the PGA Tour, and then came from behind to secure a victory over the reigning champion Hale Irwin in the Piccadilly World Match Play Championship at Wentworth Club in England.

He became the fourth Australian major champion (after Jim Ferrier, Peter Thomson and Kel Nagle) and the first to win a U.S. Open.

In the 1979 PGA Championship, at Oakland Hills, outside Detroit, Michigan, he stood on the last tee at 7 under par for his final round and leading by two, but double-bogeyed the last hole for a 65 to drop back into a playoff with Ben Crenshaw.

Graham shot a 67 in the final round to overturn a three-shot deficit to overnight leader George Burns to win by 3 strokes.

On 27 June 2004, during the final round of the Bank of America Championship on the Champions Tour, Graham collapsed over a putt on the eighth green.