Shankar's foundation said in a statement that he had suffered upper respiratory and heart problems and had undergone heart-valve replacement surgery last week, reports the Associated Press.
Indian Prime Minister Manmohan Singh also confirmed Shankar's death, calling him a "national treasure."
But it was his daughter, singer Norah Jones, who honed in on his legacy: "My Dad's music touched millions of people. He will be greatly missed by me and music lovers everywhere."
Robindra Shankar Chowdhury was born on April 7, 1920, in Varanasi (also known as Benares), India, and shot to international fame in 1965, when Beatle George Harrison studied with him. Already, Shankar, whose 1982 film score for the biopic Gandhi would earn him an Oscar nomination, had met and performed with classical violinist Yehudi Menuhin.
Other collaborators over the decades included flutist Jean-Pierre Rampal and jazz saxophonist and composer John Coltrane, who named his son Ravi.
Shankar had two children of his own: daughters Anoushka Shankar, a sitar player, and the Grammy-winning singer-songwriter Norah Jones. Her mother, concert promoter Sue Jones, and Shankar had had a long relationship.
After helping Harrison master the sitar, which resulted in the Beatles' song "Norwegian Wood," Shankar went on to influence the Rolling Stones, the Animals, the Byrds and other rock groups, and made Shankar something of a rock star himself. He performed the 1967 Monterey International Pop Festival and at Woodstock in 1969.
Evan Agostini / AP
He is survived by his wife, Sukanya Rajan, whom he married in 1989, and his two daughters, three grandchildren and four great-grandchildren.
With reporting by KEVIN O'DONNELL