The First Minister, who was interviewed by the presenter on numerous occasions, paid his respects to a broadcaster who could be fearsome.
Sir David died after becoming ill on board the Queen Elizabeth, where he was giving a speech on Saturday night.
Mr Salmond, who was most recently interviewed by Sir David in February last year when the interviewer was working for the Al Jazeera TV channel, said: "Sir David made an extraordinary contribution to current affairs, to journalism and the art of interviewing.
"He is one of the few interviewers to see his work turned into an acclaimed film, with the dramatisation of his seminal interviews with President Nixon. These showed Sir David as a man of outstanding talents who inspired many a young journalist."
Prime Minister David Cameron said of Sir Davidt, who dominated the television landscape for more than fifty years: "He was an extraordinary man, with charm, wit, talent, intelligence and warmth in equal measure. He made a huge impact on television and politics. The Nixon interviews were among the great broadcast moments, but there were many other brilliant interviews. He could be, and certainly was with me, both a friend and a fearsome interviewer."
Over a 12-year period he interviewed presidents, prime ministers and monarchs on his BBC Sunday morning political show, Breakfast With Frost.
His famous interviews with the disgraced former US president Richard Nixon were turned into a 2008 movie, Frost/Nixon, starring Michael Sheen.
He emerged as a television icon in the early Sixties with the cult satirical series That Was The Week That Was, and went on to front coast-to-coast programmes in the United States while broadcasting almost simultaneously in the UK.
Former Liberal Democrat leader Sir Menzies Campbell said: "David Frost could do you over without you realising it until it was too late. He was a peerless broadcaster."
Mr Campbell added that Sir David had an extraordinary social life but that he never stopped working. "Tony Blair said in his book Sir David was one of the best as he was so nice - by the time you answered the question you probably said something you didn't mean to."
Alastair Campbell, Tony Blair's former director of communications, added: "It's really, really sad. He was somebody you could be very close to and I would say he was a great friend to me and I hope I was to his family."
Sir David wrote 17 books, produced several films and started two television networks, London Weekend Television and TV-am.
Esther Rantzen said the key was Sir David's "capacity to elicit memorable, sometimes historically significant quotes from all the movers and shakers or our time ."
Obituary: Page 20