Their mother, Shiva, was also born in Besançon zoo eight years ago.
Their father, Tejas, was born in Bristol zoo in the UK five years ago.
The Asiatic lion is on the International Union for Conservation of Nature's red list of endangered species and Besançon zoo joined a European programme for breeding them 14 years ago.
Compared to the 100,000 African lions, there are only about 350 Asiatic lions (Panthera leo persica) living in the wild today - all of them in the Gir forest in India - and there are 100 in zoos in Europe, according to the museum.
"These births give us hope for the survival of this threatened species," the zoo's vet, Mélanie Berthet, told the AFP news agency. "Today of the 56 zoological establishments in the world that have Asiatic lions only eight of them have recorded births."
One threat to the cubs' survival could have been their own father.
Shiva was separated from Tejas 10 days before the birth and she and her cubs are being kept apart from him at the moment, although near enough for him to see and smell them.
"We will establish contact within a month," Berthet said. "After having fed them up so that he doesn't think of them as prey."
Tejas and Shiva had another cub at Besançon last year.