Ano said security forces had killed a key Abu Sayyaf leader known by his alias Abu Rahmi.
"Abu Rami is a young, aggressive and upcoming leader... who has the potential of being the next leader of the Abu Sayyaf".
"This is a very big accomplishment and a big blow to the Abu Sayyaf". "If they have further plans to kidnap innocent people somewhere, they will now have to think twice".
Askali was leading a band that sailed northward for almost 800km from an Abu Sayyaf base in Indanan town, Sulu province, to Bohol.
The Abu Sayyaf's foray into Bohol suggests that navy patrols in waters separating Sulu and Sabah are forcing the group to venture deep into the heartland of central Philippines, where millions of tourists flock to for beach and diving vacations.
The military spokesman said the goal was to kill or capture by Wednesday the remaining infiltrators and contain the fighting outside the town of Inabanga, about 80 kilometres (50 miles) from the tourist section of Bohol.
Bohol island is about an hour by boat from Cebu province, a trade and tourism centre, which has hosted some of the meetings of the Association of Southeast Asian Nations, the regional bloc the Philippines is leading this year.
Philippine intelligence agencies got wind of the plot last week, and alerted the military and foreign counterpart agencies, though authorities lost track of the gunmen in the open seas.
The gunmen traveled on motorboats along a river to Inabanga's Napo village, where government forces assaulted them, military spokesman Col. Edgard Arevalo said, adding that troops recovered four rifles and a homemade bomb from the dead gunmen.
The U.S. Embassy in the Philippines issued a warning last week that it had received "unsubstantiated yet credible information" that terrorist groups may attempt kidnappings in Bohol and Cebu provinces.
The United States and the Philippines both list Abu Sayyaf as a terrorist organization for bombings, kidnappings for ransom and beheadings. In 2001, they sailed as far as western Palawan province, where they seized 20 people, including three Americans, from a resort, two of whom were later killed.
The two Canadian men were beheaded past year after their families and their government failed to pay ransom, while the Norwegian man and the Filipina were later freed unharmed.
Ano said the gang, which arrived on three boats, had planned to acclimatise in the area and send scouts into resorts to scope out kidnapping targets.
The militant group is believed to still be holding at least 26 more hostages, including 12 Vietnamese sailors, seven Indonesians and a Dutch man who was kidnapped in 2012.