The tattered garment -- almost 80 years old -- was sun-faded, showed signs of "insect damage" and had a torn peak.
Auction house Bonhams said Bradman wore the cap during India's 1947-48 tour of Australia, his last Test series on home soil.
In an auction lasting 10 minutes, a flurry of bidding pushed the price from a starting point of $160,000 to a winning offer of $250,000 (Aus$390,000).
The total cost was $310,000 once "buyer's premium" fees were tacked on.
Bonhams said it was "the only known baggy green" worn by Bradman during the series, in which he scored 715 runs in six innings at an average of 178.75, with three centuries and a double-hundred.
Australia's cricketers are awarded the dark green woollen caps before Test debuts and they are revered by players and fans alike, often the more battered the better.
A different "baggy green" worn by Bradman during his Test debut in 1928 fetched $290,000 when it went under the hammer in 2020.
That was far less than the $650,000 paid for Shane Warne's baggy green when he put it up for sale to help Australian bushfire victims earlier that year.
Bradman retired with an all-time-high Test batting average of 99.94 and has been described by cricket authority Wisden as the greatest to "have ever graced the gentleman's game".
He died in 2001 aged 92.
