The ceremony came hours after a plane chartered by the Indian government brought her body back to the city.
The 23-year-old medical student died in a Singapore hospital where she was being treated for severe injuries.
The attack sparked two weeks of protests about gender attitudes in India, and calls for changes to laws on rape and violence against women.
Indian Prime Minister Manmohan Singh and the head of India’s governing Congress party Sonia Gandhi were at the airport when the plane landed.
A convoy carrying a gold-coloured coffin and the victim’s parents then drove towards the Janakpuri district of Delhi where she had been living.
The government has been heavily criticised for its response to the attack and remains anxious about a backlash, with police still cordoning off the heart of the capital to prevent demonstrations.
Mrs Gandhi has promised to fight what she called India’s shameful social mindsets that lie behind such crimes.
Six men arrested for the 16 December rape have been charged with murder. If convicted, they face the death penalty.