Friday, bringing to power a man whose controversial past at one point led the United States to deny him a visa.
Official results were expected late Friday.
Viewed as pro-business, Modi, 63, has pledged reforms to revive the nation's flagging economy.
But his past is not without controversy. Throughout his campaign, his relationship with the country's huge Muslim minority came under scrutiny.
In 2002, Gujarat state was wracked with anti-Muslim violence, in which more than 1,000 people, mostly Muslims, were killed.
Modi, the state's chief minister, was criticized for not doing enough to halt the violence, but a Supreme Court-ordered investigation absolved him of blame last year.
Manmohan Singh, India's outgoing Prime Minister, will tender his resignation to the nation's President on Saturday, said Singh's spokesman, Pankaj Pachauri. The Prime Minister's official Twitter account said Singh had called Modi to congratulate him on his "party's victory."
Analysts predict his arrival in India's top office will bring a marked change in direction for the world's most populous democracy, a nation whose modern character has been defined by the defeated Indian National Congress Party, which has been dominant since the country's independence in 1947.
Modi's victory had long been anticipated, as polls indicated a slump in support for the ruling Congress Party, which has been dogged by high-profile corruption scandals amid stubborn inflation and a slowed economy.
Congress Party spokesman Randeep Surjewala told CNN, "Trends indicate a victory for the opposition alliance.
"We bow before the wishes of the people of India with all humility. We will continue to play the role assigned to us. We will try with greater vigor and determination to work with the large populace of this country."
Modi's relationship with the rest of the world
The United States denied Modi a visa over the anti-Muslim violence in 2005, suggesting a strained relationship between the U.S. and India's next Prime Minister.
Before the election, U.S. State Department had not said what it would do when Modi applies for a visa in the future, but reiterated that India is an important partner.
"We don't talk about visa applications," State Department spokeswoman Jen Psaki said this week. "We're looking forward to working with the new Indian government when they're elected."
On Friday, three State Department officials told CNN that Modi will be given a visa to the United States once he takes office and forms a government.
U.S. Secretary of State John Kerry congratulated Modi, saying in a tweet that he looks forward to "growing shared prosperity."
The tensions between Modi and the United States in the past could have an impact on relations during his term, said Arati Jerath, an analyst and journalist in India.
"There is a feeling that Narendra Modi will be much more pro-China than pro-U.S., and that could be rooted to the fact that he's had this tension with the United States over his visa, whereas the Chinese laid out the red carpet for him," Jerath said.
Modi's ascent to the national stage
Celebrations broke out as updates from the five-week-long election were released throughout the day. Modi's supporters sang, danced, played music, threw flowers and even brought elephants into the mix as initial results indicated a huge lead for the BJP. Supporters celebrated outside the party's office and in the streets in Gujarat, where Modi has served as chief minister since 2001.
He tweeted: "Good days are here to come."