According to media reports, Svensson had been diving with a group and an instructor in the Stockholm archipelago when he suddenly disappeared.
The married father of two had been found badly injured on the sea bed and had been transported by helicopter to a Stockholm hospital, where he died shortly before midnight on Saturday.
Swedish police are investigating the accident.
Svensson, a genre-defying musician who reached audiences far beyond the traditional jazz crowd, was “the most important figure in jazz in this decade,” said Burkhard Hopper, manager of the Esbjörn Svensson Trio (E.S.T.).
“By all standards he was one of the biggest European acts. He was a jazz musician with pop status,” Hopper told AFP, comparing his influence to that of Miles Davis.
E.S.T., formed in 1993 by pianist and composer Svensson, drummer Magnus Öström and bass player Dan Berglund, have released a dozen records that have enjoyed critical acclaim and commercial success in Europe and the United States.
Their 2001 record “Good Morning Susie Soho” was voted best album of the year by Britain’s Jazzwise Magazine, while their 2002 “Strange Place for Snow” won numerous awards, including the German Jazz Award, the Guinness Jazz in Europe Award, the Victoire de la Musique and the BBC Jazz Award.
The group, which has won a string of other Swedish and international awards and was in 2006 the first European jazz band to grace the cover of US jazz bible Down Beat, released its last album “e.s.t. live in hamburg” at the end of 2007.
“Esbjörn Svensson was extremely important for jazz in Sweden, and even abroad,” said Bosse Persson, the head of the Stockholm jazz festival, where E.S.T. had been scheduled to perform in mid-July.
“He virtually revitalized jazz and created new audiences and inspired musicians,” Persson told the Svenska Dagbladet daily.
Bengt Säve-Söderbergh, the head of Sweden’s national jazz association, agreed.
“He will be deeply missed. There were so many people who liked his music and those who got to know him liked him as a person. He was a warm human being,” Säve-Söderbergh told Svenska Dagbladet.
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