The grant of A$1.3m (£550,000) comes after decades of speculation, debate and official inquiries as to how the light cruiser went down in the Indian Ocean after a chance encounter with the smaller and less armed German raider Kormoran.
Prime Minister John Howard said: "The finding of the HMAS Sydney would close a significant chapter in Australia's wartime history and bring a long-awaited closure to the suffering of the families, as well as allowing the proper recognition of the sacrifice made by the 645 crew," he said.
The change of heart came after the shipwreck hunter who in 2001 found the remains of the German battleship Bismarck and the British battlecruiser Hood in the Atlantic uncovered new evidence about the two ships' possible locations.
David Mearns, who is based in Midhurst, West Sussex, found forgotten documents taken from survivors of the Kormoran in the Royal Navy's historical records branch near Trafalgar Square in London.
Together with freshly decoded accounts made by the Germans while they were POWs in Australia and first-hand interviews with surviving Kormoran crew he has defined a large but economically sized search area in deep water about 200 miles offshore.
"I think that's a very, very deep scar in Australia that's never healed and I have had 10 times more interest from a country I don't live in about my work on HMAS Sydney than I had with HMS Hood," Mr Mearns said.
The issue has been contested for years with claims that a Japanese submarine may have been involved in the action, which happened just before Pearl Harbour, and that the Germans lied and shot all the survivors.
Barbara Craill, the daughter of one of the Sydney's crew, is sceptical and disappointed in the government's offer. "$1.3m towards it is a start, but it is still a long way off and there's a great many people who don't believe it's where this company says it is, that the Germans lied," she said.
The plan is to find the wreck of the Kormoran first and then search the bearing taken by the heavily damaged Australian ship. The only thing ever recovered from it was a small life raft damaged by gunfire. A seismic survey ship is expected to start the search next year. The sites of the wrecks will be marked but respected as war graves.Reuse content