Greig was 66. He breathed his last at a hospital here after being brought in a "critical condition".
"He was rushed into St Vincent's Hospital. The staff of the emergency department worked on Mr. Greig to no avail." hospital spokesman David Faktort said.
In his 58-match Test career, Greig scored 3,599 runs at an average of 40.43 and picked up 141 wickets. He slammed eight centuries during his Test career. In 22 ODIs that he played, Greig scored 269 runs and managed 19 wickets.
He was diagnosed with lung cancer in October this year after undergoing treatment for what was initially thought to be bronchitis in May.
He underwent tests after the World Twenty20 Championships in Sri Lanka and it was revealed that there was a small lesion at the base of his right lung.
Greig's son Mark had stated that his father's cancer had progressed to "stage four".
While commentating during the coverage of the first Australia-South Africa in November, Greig spoke about the disease.
"It's not good. The truth is I've got lung cancer. Now it's a case of what they can do," Greig had said.
Born in Queenstown, South Africa, Greig qualified to play for England due to his Scottish parentage. His father was a Scot.
Greig was a leading international all-rounder for England. Considered a controversial figure, he helped Kerry Packer start the World Series Cricket by signing up many English as well as some West Indian and Pakistani cricketers.
The move to start the World Series Cricket with Packer ended up costing him England's captaincy.
The best performance of Greig's captaincy career came in 1976-77, when England toured India for a five-Test series.
The team had not won there for 15 years but went on to score one of their most convincing triumphs when they clinched the first three Tests by huge margins.
Greig turned into a successful commentator following the end of his playing career in 1977 and was forthright with his views. He was one of the most bitter critics of BCCI's opposition to the Decision Review System.
A member of Australia's Channel Nine commentary team, he was described as a "beloved" figure by the network.
"Tony Greig is a name synonymous with Australian cricket -- from his playing days as the English captain we loved to hate, to his senior role in the revolution of World Series Cricket, his infamous car keys in the pitch reports and more than three decades of colourful and expert commentary," Channel Nine said.