"This is a record price for a cricket bat and there were several bidders, all of them within Australia, hoping to win the auction," a spokeswoman for Melbourne's Leski Auctions said.
Bradman made 18 and 1 in the Test at Brisbane in 1928-29, and was dropped for the next game for the only time in his career, being relegated to 12th man.
He donated the bat to a competition run by the Sun newspaper in Sydney to help raise money to endow a Don Bradman Cot for the Royal Alexandra Hospital for Children.
There had been concerns that the bat would be bought by an overseas bidder. On the eve of the sale, Charles Leski, the auctioneers' managing director, said there had been five serious inquiries from India.
"From our point of view as auctioneers we should not mind where it goes, but as an Australian this strikes me as an important Australian icon in danger of leaving the country," he said. "I don't mind if a fanatic or museum in India eventually gets it, but I wonder how some of our cricket personalities would explain that."