He is credited with the BJP's win in the recent election in the politically crucial state of Uttar Pradesh.
He replaces Rajnath Singh, who is the home minister in the new government.
A controversial politician, Mr Shah is accused of sanctioning the killing of a Muslim civilian in 2005, when he was the home minister of Gujarat state.
In 2010, Mr Shah resigned after he was charged with murder and kidnapping of Sohrabuddin Sheikh and arrested in connection with the killing.
He spent more than three months in jail after which he was released on bail. Mr Shah denies the charges.
Mr Shah, a general secretary of the BJP, was chosen its new president by the parliamentary board consisting of top party leadership on Wednesday.
Announcing his appointment, the outgoing chief Rajnath Singh said Mr Shah was chosen unanimously by all members of the board.
The 49-year-old is reported to be one of the youngest presidents of the party.
He has a reputation for being a good organiser - in the run up to the general election, he was appointed to head the BJP's campaign in the most populous state of Uttar Pradesh where he helped the party win an unprecedented 71 of the 80 seats for the party.
During the campaign, the Election Commission barred him from addressing rallies after finding him guilty of giving "hate speeches" against the Muslim community.
The ban was lifted after Mr Shah apologised and promised not to "use abusive or derogatory language".
A long-time member of the Hindu nationalist Rashtriya Swayamsevak Sangh (RSS), the ideological fountainhead of the BJP, Mr Shah has known Mr Modi for more than three decades.
Correspondents say his appointment to the top post will give Mr Modi complete control over the party and the government.

Opposition lawmakers tried to block discussions on the introduction of manual vote-counting for next year's presidential election.
The government wants a back-up to the electronic system but the opposition says it is a way of rigging the poll.
Kenyan politics is known to be highly partisan and this is not the first physical fight.
In 2014, four lawmakers were assaulted and one had his shirt torn.
Africa Live: BBC news updates
The live video feed from parliament was cut after the fight broke out on Tuesday evening and journalists were ordered out of the press gallery.
After an earlier highly charged discussion over the proposed changes to the electoral law, opposition MP Millie Odhiambo attacked President Uhuru Kenyatta, calling him "extremely stupid".
Mr Kenyatta gave his initial response to that attack at an event to usher in the festive season, saying "some idiots continue to insult me" but added that it was part of their freedom.
But Mr Kenyatta seems to have now ended his war of words with Ms Odhiambo, as a local media station on Wednesday shared a picture of a Christmas card from the president addressed to the MP.

Mr Ghani said IS was "not an Afghan phenomenon" and its atrocities had "alienated the people".
Amit Shah, a close aide of Prime Minister Narendra Modi, has been appointed the chief of India's governing Bharatiya Janata Party (BJP).
[NEXT_CONCEPT]
A parliament session in Kenya has descended into chaos, with rival MPs trading punches.
[NEXT_CONCEPT]
Afghan President Ashraf Ghani has promised to "bury" the so-called Islamic State (IS) militant group, whose local offshoot has clashed with government forces and Taliban fighters, in an exclusive interview.