After monitoring the vessels overnight, a Royal Marine team launched a "well-planned operation" at dawn, it said.
Ten Somalis surrendered and the two smaller boats were destroyed.
Commander Simon Huntington, commanding officer of Devonport-based HMS Chatham, said he was "extremely pleased" the warship had "successfully disrupted a pirate attack group operating in the Somali Basin and prevented them from mounting attacks against merchant shipping".
He said: "This clearly demonstrates Nato's determination and commitment to continue the fight against piracy in the region."
Nato said prior to boarding the boats, the suspected pirates had been observed throwing items, including their weapons and other piracy related equipment, into the sea.
When the Royal Marine team boarded the larger craft, 10 Somalis and a large amount of fuel were found on board.
The two smaller boats had been fitted with powerful outboard engines and also contained a considerable amount of fuel.
Once separated from the larger craft by the Royal Marines team, the warship and its Lynx helicopter destroyed the smaller craft so the suspected pirates could not continue with their mission.
Nato said the 10 Somalis were left with only enough fuel in the larger vessel to return to Somalia.
The search was coordinated with a EU Naval Force Maritime Patrol Aircraft, operating out of the Seychelles.bbc