The procedure, using hearts that had stopped beating, has been described as a "paradigm shift" that will herald a major increase in the pool of hearts available for transplantation.
It is predicted the breakthrough will save the lives of 30 per cent more heart transplant patients.
Until now, transplant units have relied solely on still- beating donor hearts from brain-dead patients.
But the team at the lung transplant unit of St Vincent's Hospital here announced they had transplanted three heart failure patients using donor hearts that had stopped beating for 20 minutes.
Cardiothoracic surgeon Kumud Dhital, who performed the transplants with hearts donated after circulatory death (DCD), said he "kicked the air" when the first surgery was successful.
It was possible thanks to new technology, he said. "The incredible development of the preservation solution with this technology of being able to preserve the heart, resuscitate it and to assess the function of the heart has made this possible, he said.
Hearts are the only organ that is not used after the heart has stopped beating - known as donation after circulatory death.
The novel technique used in Sydney involved taking a heart that had stopped beating and reviving it in a machine known as a "heart-in-a-box".
The heart is kept warm, the heartbeat is restored and a nourishing fluid helps reduce damage to the heart muscle.
Gribilas, 57, who was suffering from congenital heart failure. She had the surgery more than two months ago.
"Now I'm a different person altogether," she said. "I feel like I'm 40 years old - I'm very lucky."
Prof Peter MacDonald, head of St Vincent's heart transplant unit, said: "This breakthrough represents a major inroad to reducing the shortage of donor organs."
It is thought the heart-in-a-box, which is being tested at sites around the world, could save up to 30 per cent more lives by increasing the number of available organs.
The breakthrough has been welcomed around the world.
The British Heart Foundation described it as a "significant development".
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