Saxophonist Sir John, 82, served as musical director to the likes of Nat King Cole and Ella Fitzgerald.
On his Twitter page, jazz singer Jamie Cullum hailed him “a genius”, adding, “Rest in Peace, sir”.
Sir John, known as Johnny, died in a London hospital on Saturday. He had been ill for several months.
His wife, the singer Dame Cleo Laine announced his death at a concert at their Buckinghamshire theatre, where she was performing with their children.
Dame Cleo (photographed with Dankworth, above), broke the news to the artists before the concert began but did not tell the audience until the finale.
Sir John was hailed by Jazzwise magazine as "one of the totemic figures of British jazz" and the UK's "first major jazz musician".
Jamie Cullum wrote: "Sir John Dankworth, a great man and one of our finest musicians and composers, has died. Rest in peace, sir."
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