Surtees, who claimed the 1964 Formula One title after winning four 500cc world championships, died at St George's Hospital, London, after being treated for an existing respiratory condition.
A family statement said Surtees passed away peacefully with his wife, Jane, and daughters Leonora and Edwina by his side.
"We deeply mourn the loss of such an incredible, kind and loving man as well as celebrate his amazing life," the statement added. "John was a loving husband, father, brother and friend. He was also one of the true greats of motorsport.
"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."
John Norman Surtees was born on 11 February 1934 in Tatsfield in Surrey, the eldest of three children. He was brought up in Shirley where his father, Jack, ran a shop selling motorbikes and sidecars. He also raced his own constructions in his spare time.
But it was the son who would find fame. John Surtees won four 500cc motorcycling titles - in 1956, 1958, 1959 and 1960 - and the F1 crown with Ferrari in 1964.
Surtees's death sparked a flood of tributes from throughout the motorsport world.
Damon Hill, another of Britain's F1 world champions, whose late father Graham competed against Surtees, tweeted: "Such a lovely man. We have lost a true great motorsport legend. RIP John. #JohnSurtees."
Valtteri Bottas, a driver with current world champions Mercedes, said: "Sad news today. It was great to meet you legend. RIP John Surtees."