The Australian driver - who was knighted in 1979 - won world titles in 1959 and 1960 and became the only F1 driver to win a world championship in a car of his own construction - the rear-engined BT19 - which he drove to the title in 1966.
The following year the Brabham team won its second successive world championship when New Zealander Denny Hulme drove the BT20 to victory.
Sir Jack won his final Grand Prix race in South Africa in 1970 before retiring from F1 at the age of 44.
He had 14 Grand Prix wins and took pole position 13 times. He also placed second 10 times and had seven third-place finishes, giving him 31 podium finishes in his career.
"It's a very sad day for all of us," Sir Jack's youngest son, David, said in a statement on the Brabham.co.uk website.
"He lived an incredible life, achieving more than anyone would ever dream of and he will continue to live on through the astounding legacy he leaves behind."
Australian Grand Prix chairman Ron Walker said Sir Jack did a lot to develop Formula One racing, and "was a great Australian...and I am sad to learn of his passing".
Sir Jack is survived by his second wife, Margaret, and his three sons Geoff, Gary, and David.
Born on April 2, 1926, in the southern Sydney suburb of Hurstville, he grew up driving and maintaining his father's fruit and vegetable delivery vehicles.
After a brief career in engineering, he joined the Australian Air Force as a flight mechanic and went on to set up his own engineering works in Sydney, later becoming a pilot.
He won four Australian championships between 1948 and 1951. In 1955, Sir Jack moved to England and teamed up with John and Charles Cooper to make his Grand Prix debut at Aintree.
He returned to Australia after he retired and his new interests included developing a farm, car dealership and aviation company.
He was also a spokesman for a major Japanese car firm and maintained his interest in the sport, visiting numerous major international races.