Hussain collaborated with musicians including George Harrison from The Beatles, Mickey Hart from The Grateful Dead and jazz musician John Handy in a career spanning decades.
He died of a lung disease in the United States, local media reported, quoting a statement from his family.
Tributes poured in with fellow musician Amjad Ali Khan -- who plays the Indian lute-like sarod -- saying Hussain would "continue to bring the house down in the heavens".
Indian Prime Minister Narendra Modi said he would be "remembered as a true genius who revolutionised the world of Indian classical music".
Modi said he "brought the tabla to the global stage, captivating millions with his unparalleled rhythm".
Hussain was born in India's financial hub Mumbai, then known as Bombay, in 1951 to Alla Rakha, a tabla maestro himself.
He moved to the United States in 1970 and later formed the fusion band "Shakti", or "Power", with British guitarist John McLaughlin.
This year, the group won the Grammy for best global music album for their record "This Moment".
The tabla, Hussain said in an interview with news agency Press Trust of India last year, was a "mate, a brother, a friend".
"I find myself at a place where I cannot imagine that I can exist without it. It motivates me to get up in the morning and say, 'hello'," he said.
