A fierce competitor, brilliant engineer and astute businessman, Brabham claimed the Formula One titles in 1959 and 1960 for Cooper Racing before going on to win a third in 1966 for the Brabham marque.
He died at his home on Australia's Gold Coast.
"It's a very sad day for all of us," his youngest son David, who also raced in Formula One, said in a statement.
"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."
A former Royal Australian Air Force mechanic, Brabham began racing midget cars on cinder tracks in Australia in 1948 before moving to Britain to pursue his career in the mid 1950s.
Brabham became the first Australian to win the Formula One title in 1959, famously pushing his car uphill to the finishing line to seal the triumph after running out of fuel on the final lap at the U.S. Grand Prix at Sebring.
After his second triumph for Cooper, Brabham set up a company with friend and fellow Australian Ron Tauranac to design and build their own cars, one of which he drove to the Formula One title in 1966 at the age of 40.
"On track he was always the toughest of tough competitors, tough sometimes to the point at which I'd wonder how could such a nice bloke out of a car grow such horns and a tail inside one," his British rival Stirling Moss recalled in the foreword to the "The Jack Brabham Story" in 2004.
"You'd always know when Jack was on a charge because he'd crouch down and almost disappear within the cockpit. Tail-out, broadsiding, showering me with gravel and tuffets from the verge.
"Dear me, you could take the Aussie out of the dirt tracks but you couldn't take the dirt tracks out of the Aussie. But the greater side of Jack's character was always his natural sportsmanship."
Nicknamed "Black Jack" for his mop of dark hair and taciturn nature, Brabham would become "Geriatric Jack" as he raced on into his 40s, his last victory coming at the 1970 South African Grand Prix in his final season when he was 43.
In total, Brabham raced in 126 grands prix, taking pole position 13 times and winning 14 races.
After retirement, Brabham sold his his team to Bernie Ecclestone, the Briton who would go on to run the sport, and returned to Australia. He was knighted for services to motor sport in 1979.
His sons Geoff, Gary, and David later forged their own careers in motorsport, while the Brabham team name remained in Formula One until the early 1990s.
"The word 'legend' is often used to describe successful sportsmen, but often it exaggerates their status. In the case of Sir Jack Brabham, however, it's entirely justified," McLaren team boss Ron Dennis, who worked on the Cooper and Brabham teams in the 1960s, said in a tribute.
"A three-time Formula One world champion, he remains the only driver to win a Formula One world championship driving a car bearing his own name - a unique achievement that will surely never be matched."