Keyboardist and vocalist Rose Bygrave joined the band in 1979, and with Warwick Harwood (lead guitar, vocals), Ian Morrison (vocals, harmonica), Carl Smith (bass), and Gary Crothall (drums), recorded the independent 12" EP Livin' on the Razor's Edge.
Goanna signed to WEA in 1981 and supported James Taylor's Australian tour. Despite reluctance from WEA, the group's defining single, "Solid Rock," was released in October 1982, a damning rock song about the European invasion of Australia. A surprise hit, it peaked at number three on the national Australian charts and would later appear on the 1989 Building Bridges album with various other artists.
Their debut full-length album, Spirit of Place, was released in December 1982 and reached number three on the national charts. An ever-changing lineup saw Harwood leave the band and backing vocalist Suzie Dickinson join soon after its release.
In May 1983, Goanna issued the single "Let the Franklin Flow" under the pseudonym Gordon Franklin and the Wilderness Ensemble in support of the Tasmanian Wilderness Society's campaign against the proposed damming of Tasmania's Franklin River. The single reached number 12 on the national charts.
Their final album, Oceania, was released in April 1985 and peaked at number 20 during May. A national tour was halted in September 1985 when Shane Howard went missing, causing $20,000 worth of bookings to be canceled. His departure signaled the end of the band. It later transpired that he had gone to South Australia, probably due to disillusionment caused by constant touring, band debts, and an unstable band lineup.