Jacobs presented peak-time radio and TV programmes such as Jukebox Jury and Pick Of The Pops, What's My Line? and Any Questions?.
He stepped down from presenting his weekly Radio 2 programme last month due to ill health.
A BBC spokesman said Jacobs died peacefully at home, surrounded by his family.
In 1984, he received the Sony Gold Award for outstanding contribution to radio and, following his investiture as CBE, was appointed deputy lieutenant and subsequently high steward for the Royal Borough of Kingston.
Bob Shennan, controller of BBC Radio 2, 6 Music, Asian Network and popular music, said: "David was a legend in broadcasting, not only for the Radio 2 audience, but for the whole population. He was a true giant of the BBC, whose career spanned seven decades on radio and television.
"His broadcast hallmarks were great taste, authority and warmth. I am sure his audience will feel they have lost a friend, as we all do here at Radio 2."
Helen Boaden, controller of BBC Radio, said: "From Jukebox Jury to Melodies For You on Radio 2, David's effortless presenting style belied his consummate professionalism.
"David was a true broadcasting legend, as well as one of the industry's best-loved personalities, and he will be sorely missed by his devoted listeners. Our deepest sympathy goes to his family."
Tony Hall, director general of the BBC, said: "I'm very sad indeed to hear the news about David.
"As a young and avid viewer of Jukebox Jury, I remember him every week scoring the hits and misses. And I was still listening to him just last month as he fronted his show The David Jacobs Collection on Radio 2.
"He was one of the great broadcast personalities, and we shall all miss him tremendously."
BBC Radio 2 will be paying tribute to Jacobs's life with a number of programmes.
It was announced in July that Jacobs would step down from his weekly Radio 2 show, which he had hosted for 16 years, after a final edition on August 4.
He had been suffering from Parkinson's disease and had been treated for liver cancer, which kept him off air for some time during the past two years, and he said it was with "sadness" that he had decided to stop.
The presenter, whose hour-long David Jacobs Collection programme was broadcast on Sunday nights, said he hoped to host occasional shows later in the year.
He said: "Over the past two years, Radio 2 has given me time to be treated for liver cancer and Parkinson's disease.
"My producer, Alan Boyd, has been a tower of strength in so many ways and I thank him profusely. We shared a love of so many records and wonderful music.
"I will not stop collecting but my sadness will be that I cannot share them with all my loyal listeners. But rest assured, I will be back from time to time."
Jacobs first became an announcer in 1944 on the BBC General Forces Programme's Navy Mixture while he served in the Royal Navy during the Second World War.
He went on to join the BBC as an announcer and newsreader and other shows over the years included Housewives' Choice and Melodies For You.
Juke Box Jury was a big hit in the 1960s on TV, and he also hosted the Eurovision Song Contest for many