Marsh played backyard cricket with his older brother Graham, who became a professional golfer and won eleven times on the European Tour. Both brothers represented WA at cricket at schoolboy level. Marsh played his first competitive match at the age of eight for the Armadale under-16s. He kept wicket from the start, but batting was his strength. At thirteen he captained the state schoolboys' team, and joined the West Perth district club.

When he debuted for West Perth's first XI he was a specialist batsman, as WA wicketkeeper Gordon Becker also represented the club. In order to further his keeping, Marsh joined the University club. Marsh made his first-class debut for WA, again as a specialist batsman, against the touring West Indies in 1968–69. He had an unusual match, scoring 0 and 104.

Marsh replaced the retired Becker for the 1969–70 season. At the time, Australia was touring India and South Africa with Brian Taber and Ray Jordon as the team's wicketkeepers. In the autumn of 1970, an Australian second team toured New Zealand with John MacLean as wicketkeeper. Therefore, Marsh was behind these players in the pecking order. However, he was a controversial selection for the first Test of the 1970–71 Ashes series, replacing Taber. His superior batting had won him the position.

The media was quick to criticise Marsh’s glovework in his early career, dubbing him “Iron Gloves” after he missed a number of catches. Even on his debut in the First Test in the 1970–71 Ashes series he took four catches in his first innings. His batting proved invaluable on a number of occasions and in the Fifth Test, he equalled the record for the highest Test innings by an Australian keeper, set by Don Tallon. The end of the innings was controversial; the captain Bill Lawry declared with Marsh eight runs short of a century so he could get an extra hour of bowling before stumps. When questioned by the press about his lost chance to make an historic century Marsh said he had gained forty runs instead of missing eight as he thought Lawry should have declared an hour earlier. Marsh later admitted that he was underprepared as a wicketkeeper, but he learned from watching his English counterpart Alan Knott. He and Knott did, however, concede the same number of byes in the series: 44.

Marsh became an integral part of the team as the side improved during the 1972 tour of England. He became the first Australian keeper to hit a century by scoring 118 in the first Test against Pakistan at Adelaide in 1972–73. He also hit 236 against the tourists for WA, the best score of his career. Playing a key role in Australia's series victories over England and the West Indies in the series of 1974–75 and 1975–76, Marsh made many acrobatic dives to catch balls delivered by Dennis Lillee and Jeff Thomson. He took 45 dismissals in those two series, including a world-record 26 catches in six Tests against the West Indies.

Marsh scored an unbeaten 110 in the second innings of the Centenary Test against England in 1977, becoming the first Australi