Two couples and a three-year-old child were aboard the yacht when it was seized on April 4 as it headed for the Kenyan coast.
The statement said the four other hostages, including the child, were "safe and sound".
Al Jazeera's Estelle Youssouffa, reporting from Paris, said the hostage killed was Florent Lemacon, the owner of the boat and the father of the child.
Lemacon had been in the cabin when the navy stormed the yacht and it was not clear if he was killed in the crossfire or deliberately shot by one of his captives.
"The French military says it's launching an investigation to find out if the hostage has been killed by a pirate or French military," Youssouffa said.
Herve Morin, France's defence minister, told a news conference on Friday that the operation came after the pirates rejected an offer to exchange a naval officer for the women and child on the boat.
"We proposed all possible solutions to the pirates in order to retrieve our fellow citizens, including an exchange of the mother and the child for an officer, which were refused," he said.
"Negotiations were leading nowhere, and the boat was approaching the coast."
Morin said Sarkozy gave the order to attack.
|Ship owners have paid tens of millions of dollars in ransoms to pirates [GALLO/GETTY]
Jean Louis Georgelin, French chief of defence staff, said: "At 3:30pm, we spotted three kidnappers on the boat deck and the order was given to neutralise them and at the same time to send a squad of eight navy riflemen.
"They were on the Tanit
30 seconds later. Two on the front, two on the back and the rest in the middle. Two pirates were immediately killed, the third fell in the sea."
France has taken a lead role in attempting to bring the piracy, which has been rampant in the busy shipping lanes of the coast of east Africa, under control.
French forces have detained at least 60 suspected pirates since April 2008, taking several of them back to France to face possible trial.
"France will never give into pirates' blackmail or to terrorism," Morin said.
Meanwhile, the Norwegian-owned MT Bow Asir tanker was freed after its captors received a ransom payment of about $2.4m from the owners of the vessel, sources said.
"The pirates on board the Norwegian ship took the ransom and now they are all in Haradheere with me," one pirate told the Reuters news agency by telephone.
The 23,000-tonne chemical tanker's operator, Salhus Shipping AS, confirmed that the vessel was freed and the 27-member crew was unharmed but declined to say if it paid any ransom or give details about negotiations with the pirates.
The ship and its crew were seized by pirates on March 26 about 400km east of the Somali coast.
In an other incident, pirates threatened to kill the captain of a US cargo ship held hostage by four Somali pirates after he jumped in the water in an attempt to escape.
Dozens of ships seized by pirates in the Indian Ocean have been released after their owners have paid tens of millions of dollars in ransoms.
Crispian Cuss, a security analyst, told Al Jazeera more attacks were expected.
"As the weather gets better, we're going to see more pirate activity. The pirates operate in very low boats, so they need calm waters."
Cuss said all kind of vessels, from small yachts to big cargo ships, could be targeted.
"We may be seeing increased investment in piracy. They [the pirates] had such a good year last year. They got some great ransoms. They are getting bigger and faster vessels, and are also expanding their reach."