Modi arrived in the Indian capital from western Gujarat state, which he has governed since 2001.
The long-time opposition leader is now preparing for his five-year term in office after his BJP secured the biggest win by a single party in 30 years.
On Friday, Modi addressed a crowd of cheering supporters, promising a better economy and vowing his government will take "everyone" in India along.
U.S. President Barack Obama phoned the incoming Indian prime minister Friday and invited him to visit Washington.
Singh leaves office
Meanwhile, outgoing Prime Minister Manmohan Singh has left office after his ruling Indian National Congress party was crushed in parliamentary elections.
In a brief address Saturday, Singh said all Indians must respect the verdict of voters who handed the opposition Bharatiya Janata Party (BJP) a landslide victory. He said India is a far stronger country than it was a decade ago.
Congress has led India for the past 10 years, but has been hit by a series of high-profile corruption scandals, high inflation and lagging economic growth.
India's five-week election wrapped up Monday, with the BJP winning the first one-party majority in parliament in 30 years. That means it can create a government without forming a coalition with regional leaders.
The Election Commission of India says a record 66 percent of the country's 815 million registered voters cast ballots.
Controversy has swirled around Modi since 2002 when Hindu-Muslim rioting in his state killed more than 1,000 people. But India's Supreme Court cleared Modi of charges that he incited the violence.
Modi has since cast himself as an able administrator and decisive leader who has energized the economy of Gujarat and holds the promise of doing the same for the rest of the country.
India's prime minister-elect, Narendra Modi was mobbed by flag-waving supporters as he arrived in New Delhi Saturday.