"Exercising powers vested in him under Article 75 (1) of the Constitution of India, President Kovind, today appointed @narendramodi to the office of Prime Minister of India," official Twitter account of Rashtrapati Bhavan posted.
The president has asked him to begin the process of forming the new government and he will soon inform him about the date of oath-taking ceremony, Modi told media.
In the general election, he led the BJP-led NDA to a landslide victory with the alliance winning 353 seats, including an unprecedented 303 by the BJP.
All its key allies, including Bihar Chief Minister and JD(U)'s Nitish Kumar, Shiv Sena's Uddhav Thackeray, LJP's Ram Vilas Paswan and Akali Dal's Parkash Singh Badal, were present as BJP president Amit Shah announced Modi's name as the alliance leader after formalities were done amid massive cheers from the MPs.
An NDA delegation then met Kovind and Modi staked his claim to form the government.
In his address to NDA members, Modi said his government will now begin "a new journey to build a new India with new energy".
Reaching out to minorities, he slammed the opposition for making them live in an "imaginary fear" and asked the MPs to win their trust and work without any discrimination, including on the basis of faith or caste.
"Our mantra should be 'sabka saath, sabka vikas and sabka vishwas' (with all, for everybody's development and having everyone's trust)," he said.
"They (minorities) cannot be handed over to them (opposition), who benefit because we keep quiet," he said, accusing his rivals of exploiting them during elections
as part of their votebank politics.
He said the poor were deceived all these years but his government managed to break through this "deception" and improved their lot, and asked the NDA MPs to similarly smash through the deception around minorities.
"Minorities were deceived, like the poor," the prime minister said.
"It would be good if their education, social and economic standards were lifted all these years," he said.
"We have to break through this deception. We have to win their trust... It is a big responsibility that we have to fulfil. Those who vote us are ours and those who did not are also ours. Even our most trenchant opponents are ours. We cannot believe in any discrimination," he said.
Invoking the 1857 War of Independence against the British rule, he said all communities had then fought together and called for replicating a similar unifying spirit for good governance now as India readies to celebrate the 75 years of Independence in 2022.
"The Constitution is supreme for us. Whatever form of worship we follow at home but outside there cannot be a bigger God for us than Bharat Mata (Mother India)," he said, likening the country's 130 crore citizens to 130 crore 'Gods' who should be served by the ruling alliance with a similar spirit.
Minorities, especially Muslims, are seen to be strongly opposed to the BJP due to their wariness over its Hindutva ideology.
"We stand for those who trusted us and also for those whose trust we have to win... We have to take everyone along to take India to new heights in the 21st century," Modi said.
With speculation being rife over who will join his Council of Ministers, Modi said he was yet to go through the details of NDA MPs and asked them to not trust media reports, adding they are aimed at creating confusion and often put out with "bad intentions".
Many "Narendra Modis" have cropped up in the country, giving out ministerial positions to MPs, he said wryly and added that newspaper reports do not make ministers.
Responsibilities will be given as per norms, he stressed.
Modi said elections
often divide and create gulfs but the 2019 polls united the people and society. There was a pro-incumbency sentiment in this election, he said.
"We ran the government for poor people between 2014-19 and I can say the poor elected the government this time," the prime minister said.
With the BJP having had an uneasy relation with some of its allies during the last five years, Modi underscored his party's commitment to take allies along and noted that he is making this comment despite the saffron party winning 303 seats, many more than the majority mark of 272.
Invoking the late BJP stalwart Atal Bihari Vajpayee for making coalition political successful, he called for the NDA to work with cohesion for the country's development and said his "naara" (slogan) for the alliance is "national ambition and regional aspirations".
Asking MPs to work with him, he said they should shun arrogance as it is the people not their personality, any caste or Modi who elected them.
He also asked MPs to avoid falling to temptation of publicity in the media.