Russian Deputy Prime Minister Dmitry Rogozin and Indian Defence Minister A.K. Antony took part in a ceremony at the Arctic port of Severodvinsk on Saturday morning, Russian news agencies reported.
Officials lowered the Russian flag on the ship and raised the flag of the Indian navy in its place. The wife of the ship's captain broke a coconut against its side.
Russia, which is India's largest arms supplier, carried out a major refit of the aircraft carrier, which joined the Soviet navy in 1987 as the Admiral Gorshkov. It has now been renamed the INS Vikramaditya after an Indian emperor.
The original deal signed in 2004 saw Russia agree to modernise the mothballed carrier for $771 million (571 million euros) by 2008. But costs ballooned to $2.3 billion, according to Indian media reports, as the deadline was regularly extended.
Russia says that the original deadline was based on an incorrect assessment of the amount of work that needed to be completed by its Sevmash shipyard, which specialises in nuclear submarines.
The ship's captain, Suraj Berry, and the deputy director of Russia's arms export agency, Igor Sevastyanov on Saturday signed the final papers handing over the ship, the ITAR-TASS news agency reported.
The carrier is expected to arrive in India in early 2014. It will have a team of Russian experts on board who will monitor and assist their Indian counterparts during the home journey.
The refurbished ship is 284 metres long and is designed to carry 30 aircraft including MiG-29K fighters. Its crew numbers 1,600 and the 44,500 ton-carrier has even been adapted to dish out Indian food.
The aircraft carrier will become India's second, intended to shore up the country's defence capabilities as it seeks to counter a military build-up by an increasingly assertive China.
Russia's long-delayed upgrade could damage its prospects as a supplier to India, the world's largest arms buyer, which has recently turned to Israel, France, Britain and the United States.
India's already depleted naval force received a major setback in August when a Russian-made submarine the INS Sindhurakshak exploded in a Mumbai dock, killing all 18 crewmen on board.