Mangeshkar, whose voice was the soundtrack to hundreds of Bollywood films, died aged 92 on Sunday.
PM Narendra Modi and stars of the entertainment industry attended the funeral, where large crowds gathered to pay their respects.
Her extraordinary career spanned more than half a century and she recorded thousands of songs in 36 languages.
Two days of national mourning will follow the funeral and the national flag will be flown at half-mast throughout the country.
Crowds gathered outside her home, where her coffin began its journey to a city park where it was publicly cremated. Mr Modi was among those who laid flowers on the pyre.
One mourner, Rajesh Kumar Ram, told the AFP news agency "her voice touches the soul of every Indian. Her songs have been with us all through our lives."