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The brass pistol, an early 18th century Continental Flintlock holster pistol with a 13-bore barrel made by Dutch gunmaker Godefroi Corbau Le Jeune, had a pre-sale estimate of A$100,000 to A$200,000.
One of a rare handful of personal effects remaining from the explorer, the gun went to a private buyer in Victoria, Australia, said Cassandra Hilber, at Leski Auctions.
"There was a lot of interest from New Zealand as well," she added.
Cook reached the coast of Australia in April 1770, the first recorded European to encounter the continent's eastern coast, after mapping the coastline of New Zealand. In August, he planted the British flag on Possession Island in northern Queensland.
It is not known if Cook was carrying the pistol when he first stepped on Australian soil.
Cook made two later exploratory expeditions to the Pacific and was killed in Hawaii in 1779.
The pistol remained in the Cook family for more than two centuries before being purchased by former Melbourne Lord Mayor Ron Walker at an auction in Edinburgh in 2003.