The 63-year-old pledged on Friday to fulfil the dreams of all of India's 1.2 billion people in an effort to allay fears that still linger among Muslims who remember bloody riots in Gujarat in 2002.
The strict vegetarian also pledged "to make the 21st century India's century".
Watch: Outgoing Indian PM Manmohan Singh addresses the nation
Modi is expected to turn the country sharply to the right after a decade of rule by the Congress party, which presided over corruption scandals and a slowdown in growth.
He made good governance and development the main focus of his campaign, deriding his Congress rival Rahul Gandhi - whose father, grandmother and great-grandfather were all PMs - as a "princeling" who had little concept of the aspirations of the Indian people.
- AAP