The Congress party, led by Rahul Ghandi, has presided over India's government for most of its history since independence in 1948. But voters ousted the party because of corruption scandals, high prices, rising unemployment, and slowing economic growth.
Although all of the 550 million votes have not yet been counted, the Congress party conceded the victory to the BJP.
"We humbly accept the verdict of the people of India. We shall continue to play with rigor the role of a constructive and meaningful opposition -- the role the people of India have assigned to us," said Congress party spokesman Randeep Singh Surjewala.
The size of the victory impressed and surprised observers, since BJP had little chance of success as of last summer. Modi has been particularly popular in India's rural regions, however, where the majority of voters live.
After two hours of counting election returns, Modi's party was assured of winning at least 272 seats in parliament -- enough to lead the government without seeking a coalition with any of India's many political parties.
At the celebration at BJP headquarters in New Delhi, supporter Surinder Singh Tiwana said, "This is a huge change for our country, a change of guard. A billion-plus people have announced their mandate in no uncertain terms. They have voted for a progressive, stable government."
Modi, 63, ran a high-voltage, exciting campaign, compared to a relatively listless effort by opponent Gandhi. A regional leader from Gujarat state, Modi's stern and disciplined reputation is a cause of worry for some people. Many Muslims in India blame him for failing to stop religious riots in 2002 in Gujarat that killed over 1,000 people; others fear political dissent will be suppressed.