Prime Minister Manmohan Singh hailed Shankar, who popularised Indian classical music around the world, as "a national treasure and global ambassador of India's cultural heritage".
"An era has passed away... The nation joins me to pay tributes to his unsurpassable genius, his art and his humility," he said.
Shankar, who had houses in California and India, was born into a high-caste Bengali Brahmin family in the Hindu holy city of Varanasi in northern India on April 7, 1920.
He taught close friend the late Beatle George Harrison to play the sitar and collaborated with him on several projects, including the ground-breaking Concert for Bangladesh in 1971 to raise awareness of the war-wracked nation.
Harrison called him "The Godfather of World Music" and Menuhin, himself widely considered one of the greatest violinists of the 20th century, compared him to Mozart.
Other devotees included jazz saxophonist John Coltrane, whom he taught Indian improvisation techniques.
Shankar, a three-time Grammy winner, was also on the bill with Janis Joplin and Jimi Hendrix at the Woodstock Festival in New York state in 1969 when 500,000 people gathered for one of the iconic cultural events of the century.
Dressed in traditional Indian clothes and seated on the floor when playing, he was lauded by the hippie generation but he expressed reservations about the excesses of Western stars and said his priorities were music, yoga and philosophy.
Shankar's wife Sukanya and his younger daughter Anoushka, 31, described him as a "husband, father, and musical soul".
"His health has been fragile for the past several years and (last) Thursday he underwent a surgery that could have potentially given him a new lease of life," they said in a statement.
"Unfortunately, despite the best efforts of the surgeons and doctors taking care of him, his body was not able to withstand the strain of the surgery. We were at his side when he passed away.
"Although it is a time for sorrow and sadness, it is also a time for all of us to give thanks."
His family and the Ravi Shankar Foundation said he had been suffering respiratory and heart problems.
The statement said that Shankar performed his last concert on November 4 in Long Beach, California, with Anoushka, who often played alongside him.
Shankar, who wrote the score for the 1982 Oscar-winning film "Gandhi", was survived by his second wife, two daughters, three grandchildren and four great-grandchildren.
His son Shubendhra, born to his first wife, died in 1992.
The night before his surgery he was informed that his latest album, "The Living Room Sessions, Part 1", had received a 2013 Grammy nomination.
"Indian classical music has lost its chief ambassador," said A.R. Rahman, the country's leading composer who wrote the score for "Slumdog Millionaire".
Sanjay Sharma, whose family made sitars for Shankar for more than 30 years, told AFP that as a client he was demanding but appreciative.
"He was the biggest innovator in music," Sharma said. "He wanted to revolutionise the sitar as an instrument. It was very challenging to work with him but every moment spent with him was God's valuable gift to our family."
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