The last rites were performed by his eldest son, the film star Amitabh Bachchan.
The 96-year-old poet was suffering from respiratory problems for some time and his condition had deteriorated in recent days.
Mourners included many Bollywood stars, top industrialists and politicians,
Always remembered
Police stationed outside the Bachchan family residence in Bombay, or Mumbai, had to use batons to beat back surging crowds.
The poet was cremated in Ruia park in the west of the city.
The BBC's Monica Chadha says that almost the entire Bombay film industry, including director Yash Chopra, film actors Govinda, Aishwarya Rai, Salman Khan, and Shah Rukh Khan, came in to give their condolences.
Prime Minister Atal Behari Vajpayee - a noted Hindi poet himself - also sent his condolences and said Hindi lovers would always remember the great poet.
Congress influence
Harivansh Rai Bachchan was born in 1907 in Allahabad in the northern state of Uttar Pradesh. He went to school in Allahabad and had his university education in Benares.
Allahabad was the unofficial headquarter of the Indian National Congress, a party fighting against British rule under the leadership of Mahatma Gandhi and Jawaharlal Nehru.
The young poet was influenced by the Congress movement and its concern for social unity against the British Raj. And that social unity was the running theme of his greatest poetry collection, 'Madhushala' (Tavern).
Published in 1935, Madhushala brought instant fame. It whipped up a literary frenzy when he recited it to huge audiences.
Madhushala is one of the most enduring works of Hindi literature. It has been translated into English and many regional Indian languages such as Bengali, Marathi and Malayalam.
It has been choreographed, and performed on stage. It was one the first pieces of Hindi poetry that was set to music, with its best selling cassettes and CDs having appeal across generations.
From 1941 to 1952 Bachchan taught English at Allahabad University and then moved to Cambridge to do his doctorate on 'W B Yeats and Occultism'.
There he became the first Indian to become a doctor of philosophy in English. He also translated some of the Shakespeare's plays into Hindi.
Nehru-Gandhi friendships
After returning from Cambridge, Bachchan served in India's foreign ministry, advising the government on the greater use of the Hindi language in official business.
There he came into close contact with the family of Prime Minister Jawaharlal Nehru. Friendships continued between the succeeding generations of the both families.
During the 1970s Bachchan began working on his four volume autobiography, which was later abridged as 'In the Afternoon of Time', by Dr. Rupert Snell of London University.
Again it was a milestone in Hindi literature, admired for its transparent honesty.
Harivansh Rai Bachchan is survived by his wife Teji Bachchan and sons, Amitabh and Ajitabh.