The deal to buy 66 aircraft, includes training, development and infrastructure, Defence Secretary Ajay Prasad said.
The announcement follows nearly 20 years of on-off negotiations between Britain and India over the contract, which has been stalled on arguments over the price.
"This has been a marathon hurdle race," said ABN Amro analyst Sandy Morris, adding that one of the reasons India had delayed awarding the contract for so long was that BAE would not offer a discount on the aircraft.
"If BAE had discounted the price of the Hawk, the contract would have probably been agreed much earlier," he said.
India, which has the world's fourth-largest air force, badly needs trainer jets.
Pilots now go straight from a basic low speed jet to ageing Russian MiG-21 fighters.
"Now that approval has come, we will be calling BAE representatives to finally settle contract terms with them, so we can move forward"
Ajay Prasad, Indian Defence Secretary
Prasad said 24 of the Hawk H-155 Y model planes would be built in Britain while the rest would be assembled by state-run Indian manufacturer Hindustan Aeronautics Limited.
Rolls-Royce Plc supplies engines for Hawks. "Now that approval has come, we will be calling BAE representatives to finally settle contract terms with them, so we can move forward," he said, adding that pilots would go to Britain to train on other Hawks until India got its own planes.
The proposal to buy the advanced trainer jets was cleared at a meeting of the cabinet, which has been under mounting pressure to improve the air force's safety record.
The Hawk has long been the frontrunner for the deal since the Indian Air Force formally proposed a trainer plane in 1985.
Indian air force officials say at least 170 MiGs have been lost and more than 40 pilots killed in accidents in the past decade, particularly because pilots have not been trained adequately.
The Hawk is the mainstay trainer for the US Navy and Britain's Royal Air Force, which placed a top-up order for 44 in July. Australia, Bahrain and South Africa are also recent Hawk customers.