The handover was celebrated by the group.
The sale is a boost to Mr Modi, who had been wanting to sell the government's entire interest in the airline. Since Mr Modi came to power, it has been the biggest disinvestment in government-owned assets.
Despite ambitious targets, the government has been unable to sell its stake in unprofitable companies.
prized slots at London's Heathrow airport, a fleet of more than 130 planes, and thousands of trained pilots and crew are some of the assets Air India has.
Vistara, a full service carrier in partnership with Singapore Airlines, and AirAsia India, a budget airline in partnership with Malaysia, are two of the airlines run by the company.
The merged Air India and Indian Airlines had been making losses. It was operational because of taxpayer-funded rescues.
The airline was losing money every day, according to the government.
The airline's management cited rising aviation fuel prices and airport usage charges as well as competition from low-cost carriers, a weakened rupee and the interest burden for its poor financial performance.
Air India suffered for its inconsistent service standards, low aircraft utilization, dismal on-time performance, antiquated productivity norms, lack of revenue generation skills and unsatisfactory public perception according to Jitender Bhargava, a former executive director of the airline.
The airline has several valuable assets, including millions of dollars worth of prime real estate, which has attracted buyers.
The new buyer will have control of the airline's 4,400 domestic and 1,800 international landing and parking slots, as well as 900 slots at airports overseas. Its international operations bring in more than two-thirds of its revenues.
In March last year, the aviation ministry said its fixed assets were worth more than $600 million.
Air India has more than 40,000 pieces of art and collectibles, including an ashtray designed and gifted by the Spanish surrealist artist Salvador Dali in the 1960s. The baby elephant was flown to Spain.
Analysts say the Indian market is vastly underserved and that Air India is a good prospect for the group.