"Our dear Rishi Kapoor passed away peacefully... today after a two-year battle with leukemia," his family said in a statement.
Fellow-actor Amitabh Bachchan was among the first to mourn his death, tweeting: "I am destroyed".
Indian Prime Minister Narendra Modi also paid tribute, calling him "a powerhouse of talent", while Bollywood superstar Aamir Khan hailed him as "one of the greats".
Kapoor was cremated Thursday at a private ceremony in Mumbai under heavy security, with his family urging fans to follow India's restrictions on public gatherings to prevent the spread of coronavirus.
Born September 4, 1952, into the prolific Kapoor dynasty - which has produced four generations of actors including his son, Ranbir - he made his debut in the 1970's epic "Mera Naam Joker" ("My Name is Joker").