The veteran BBC interviewer died last night on the Queen Elizabeth cruise ship, where he was giving a speech.
Known for incisive interviews with the leading figures of his time - and perhaps most famously disgraced US president Richard Nixon, Sir David spent more than 50 years as a television star.
His family said in a statement: “Sir David Frost died of a heart attack last night aboard the Queen Elizabeth where he was giving a speech.
“His family is devastated and have asked for privacy during this difficult time. A family funeral will be held in the near future and details of a memorial service will be announced in due course.”
Frost was a familiar figure in Yorkshire, where his television series The Wilson Interviews and later Through The Keyhole were recorded. But it was at BBC TV in London where he made his name, first with the satirical series That Was The Week That Was and then the comedy revue The Frost Report.
He went on to create London Weekend Television in 1968 and the breakfast station TV-am in the early Eighties.
David Cameron was quick to pay tribute and described Sir David as “an extraordinary man - with charm, wit, talent, intelligence and warmth in equal measure” who had “made a huge impact on television and politics”.
“The Nixon interviews were among the great broadcast moments - but there were many other brilliant interviews,” the Prime Minister said.
“He could be - and certainly was with me - both a friend and a fearsome interviewer.”
Actor and comedian Stephen Fry said he had spoken to Sir David only on Friday and he had “sounded so well” and was “excited about a house move, full of plans”.
Former prime minister Tony Blair referred to Sir David as a “huge figure in broadcasting, a great professional and a good friend”.
“He had an extraordinary ability to draw out the interviewee, knew exactly where the real story lay and how to get at it, and was also a thoroughly kind and good natured man,” Mr Blair said.
“Being interviewed by him was always a pleasure but also you knew that there would be multiple stories the next day arising from it.”
Former Lib Dem leader Sir Menzies Campbell also paid homage, saying: “To be interviewed by David Frost was never a chore, even when trying to defend the indefensible.
“But his scrupulous and disarming politeness hid a mind like a vice. David Frost could do you over without you realising it until it was too late. He was a peerless broadcaster.”
Sir David’s award-winning interview style was considered non-aggressive, affable and effusive, but he had a talent for extracting intriguing information and revealing reactions from his subjects.
His roster of interviewees included virtually every US president and British prime minister during his working life.
During his series of five interviews with Nixon in 1977, the notoriously slippery former president known as “Tricky Dicky” dramatically admitted that he had “let down the country”.
His appeal to American audiences saw him become one of the Concorde’s most assiduous users, and he claimed to have been on the supersonic plane “somewhere between 300 and 500 times”.
Other historic moments in his career included a tense interview with Margaret Thatcher over the sinking of the Argentine warship the Belgrano during the Falklands conflict in which he suddenly introduced the word “bonkers”.
He was also the last person to interview Mohammad Reza Pahlavi, the last Shah of Iran.
Outside world affairs, his roster ranged from Orson Welles, Tennessee Williams, Noel Coward, Peter Ustinov, Woody Allen, Muhammad Ali, the Beatles, Clint Eastwood, Sir Anthony Hopkins, Sir John Gielgud, Norman Mailer, Warren Beatty and many more.
Born on April 7, 1939, the son of a Methodist preacher, at Tenterden, Kent, he was educated at Gillingham Grammar School, Wellingborough Grammar School and Gonville and Caius College, Cambridge.
His big break came when he co-created and hosted satirical show That Was The Week Was in the early 1960s.
Another of his early programmes, The Frost Report, effectively launched John Cleese, Ronnie Barker and Ronnie Corbett on their subsequent glittering careers.
In more recent times, he had hosted Breakfast with Frost on Sunday mornings (1993-2005) and panel game show Through The Keyhole (1987-2008).
He was currently working for Al Jazeera English and had recently interviewed Chilean novelist Isabel Allende and F1 driver Lewis Hamilton.
Peter Fincham, director of television at ITV, said Sir David was the “epitome of old school charm” and “one of the giants of television”.
Mr Fincham said: “He was a major presence on screen for five decades, able to switch effortlessly from light entertainment to interviewing world leaders. And he was the most courteous and generous man you could hope to meet, always making it seem that it was his great good fortune to know you, rather than vice versa.”