Tributes have poured in since Jaffrey suffered a brain haemorrhage in London.
Jaffrey starred in such acclaimed Indian films as Satyajit Ray's Shatranj ke Khiladi (The Chess Players) and more than 100 Bollywood productions.
He also appeared in A Passage To India, Gandhi and The Man Who Would Be King, and earned a Bafta nomination for his role in 1985's My Beautiful Laundrette.
The news of his death was announced on Sunday by his niece Shaheen Aggarwal on Facebook.
She wrote: "Today, a generation of Jaffreys has passed away. Saeed Jaffrey has joined his brothers and sister and is rejoicing in the lap of his Heavenly Father, eternally."
The actor collapsed at his London home from a brain haemorrhage and never regained consciousness, according to a statement from Jaffrey Associates.
Born in Punjab in India in 1929, Jaffrey began his career as a theatre actor and was the first Indian to tour Shakespeare across the United States and to star in a major Broadway role, according to Jaffrey Associates.
He became a household name in the UK after appearing in numerous TV dramas including Tandoori Nights, The Far Pavilions and Gangsters.
His best-known roles included the Nawab of Mirat in the landmark 1980s series The Jewel In The Crown and Ravi Desai in the soap opera Coronation Street.
On the big screen, he starred opposite Michael Caine and Sean Connery in 1975's The Man Who Would Be King and played Patel in Sir Richard Attenborough's Oscar-winning Gandhi in 1982.
Sir David Lean cast him as Hamidullah in 1984 epic A Passage to India, and Jaffrey was nominated for the best supporting actor Bafta for playing the laundrette-owning Nasser in My Beautiful Laundrette, which starred Daniel Day-Lewis.
Jaffrey also wrote and starred in dozens of radio plays, winning the Prix Italia for his appearance alongside Sir Michael Redgrave in BBC Radio Four's The Pump.
He was awarded the Order of the British Empire (OBE) in 1995 for his contributions to drama.
The actor was married to actress-author Madhur Jaffrey with whom he had three children. They divorced in 1966.
-BBC