The pirates, along with their arms and equipment, would be handed over to the appropriate authorities, it said, adding that the ship would return to patrolling the Gulf of Aden.
Last month, the navy said it had sunk a pirate "mother vessel" off Somalia in a battle that started after the Indian ship INS Tabar came under attack from armed men on board the boat.
But the International Maritime Bureau later said that the vessel destroyed by the Indian navy was actually a Thai fishing trawler that had been seized by pirates off the coast of Yemen.
Somali pirates have carried out about 100 attacks in the Gulf of Aden and Indian Ocean since the start of this year.
They still hold at least 14 foreign vessels and more than 300 crew members to ransom.