Jack Brabham died at his home on the Gold Coast on Monday morning.
His youngest son David said in a statement: "It's a very sad day for all of us. My father passed away peacefully at home at the age of 88 this morning.
"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."
Brabham won the F1 world title in 1959, 1960 and 1966 and was constructors' champion in 1966 and 1967.
He won his first two crowns in the Cooper Climax, but with his third became the only driver in history to win a World Championship in a car he himself had manufactured - the Brabham BT19.
The first driver to be knighted for services to motorsport, Brabham was a skilled and innovative engineer as well as driver. He was responsible, with compatriot Ron Tauranac, for more than 500 Brabham racing cars being build at the team's Surrey base in the 1960s.
The Brabham name continued in F1 for 30 years.
Brabham, whose final win came in the 1970 South African Grand Prix, is survived by his wife, Lady Margaret, and sons Geoff, Gary and David and their families.
The Australian Grand Prix paid tribute to Brabham, saying in a statement he would "forever be remembered as one of motorsport's most influential figures, not only in Australia but around the world".
The chairman of the Australian Grand Prix corporation, Ronald J Walker, and its chief executive, Andrew Westacott, said on http://www.grandprix.com.au: "The contributions that Sir Jack made to the sport as well as the Australian Grand Prix will never be forgotten and his legacy will continue to resonate amongst drivers and fans alike."
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