Surtees, who won the F1 title with Ferrari in 1964 to add to his 500cc motorcycle world titles from 1956, 1958, 1959 and 1960, “passed away peacefully this afternoon” his family announced on Friday.
His wife Jane and daughters Leonora and Edwina were by his side.
Their statement added: “It is with great sadness that we announce the death of our husband and father, John Surtees CBE.
“John, 83, was admitted to St George’s Hospital, London in February with an existing respiratory condition and after a short period in intensive care he passed away peacefully this afternoon. His wife, Jane and daughters, Leonora and Edwina were by his side.
“John was a loving husband, father, brother and friend. He was also one of the true greats of motorsport and continued to work tirelessly up until recently with The Henry Surtees Foundation and Buckmore Park Kart Circuit.
“We deeply mourn the loss of such an incredible, kind and loving man as well as celebrate his amazing life. He has set a very real example of someone who kept pushing himself at his peak and one who continued fighting until the very end.
“We would like to thank all the staff at St George’s Hospital and The East Surrey Hospital for their professionalism and support during this difficult time for us. Thank you also to all of those who have sent their kind messages in recent weeks. Funeral details will be announced in due course.”
During a motorcycle career which saw him race for Norton and MV Augusta, he also became the first man to win the Senior TT on the Isle of Man three years in succession.
He started over 100 F1 grands prix, winning six, in a career which took him to Ferrari, Cooper and Honda, as well as starting a team which bore his name.
In 2009 his son Henry was killed in a Formula Two race at Brands Hatch, aged 18, when he was struck on the head by a wheel which had become detached from another car.
Ferrari posted a black and white picture of Surtees posing next to team founder Enzo Ferrari on Twitter, with the message: “John Surtees, Motorsport legend and 1964 World Champion with Ferrari, passed away. Our thoughts are with his family.”
Damon Hill, the 1996 F1 champion, tweeted “Such a lovely man. We have lost a true great motorsport legend. RIP John” and there was also a message on the official MotoGP account which read: “Very sorry to hear that John Surtees, the only person to win both #MotoGP & #F1 World Championships, has passed away at the age of 83.”
The Henry Surtees Award was launched in his memory by the family in 2010 to be awarded annually for the most outstanding performance by a rising motor racer.
John Surtees was awarded the CBE in the 2016 New Year Honours for services to motorsport.