Major Nasir Khan, another military spokesmen, said that jets had bombed positions in mountains in the Babaji Kandao area of Buner.
Casualty figures were not immediately known, though Abbas said that 10 soldiers and up to 75 fighters had been killed.
In an FM radio broadcast made from Swat valley, the Taliban hit out at the government of President Asif Ali Zardari.
It said his Pakistan People's Party and its ally Awami National Party, which governs the North West Frontier Province (NWFP), had betrayed the people of Swat and would now be targeted.
Pro-Taliban groups in other parts of NWFP have, however, not yet voiced their strong support for the Taliban in Swat.
Buner, Lower Dir and Swat are covered by the Malakand peace deal, which allows the Taliban to enforce their strict interpretation of sharia, or Islamic law, in the region in return for a truce.
The government offensive is straining the already shaky peace deal, which was widely criticised in the West as a surrender by Pakistan's government to expanding Taliban control.
Sufi Muhammad, the local religious leader who has been holding peace talks with the central government, has suspended dialogue with Islamabad as a result of the army assault.
Before the military launched its Buner offensive, Rehman Malik, Pakistan's interior minister, gave warning to Taliban fighters to vacate the area.
"We are not going to spare them," he said.
The US welcomed the Pakistani military operation and urged Islamabad to remain on the offensive.
"The key is to sustain these operations at this tempo and to keep the militants on their heels and to, ultimately, defeat them," Geoff Morrell, the Pentagon press secretary, said.
In Mingora, Swat's main town, residents said that Taliban fighters had taken up positions in the main market and posters placed on the walls of buildings warned journalists over their coverage of the Taliban.
"They should mend their ways, otherwise they will be responsible for the consequences," the posters said.
Flight of locals
The government and rights groups say that about 33,000 people have left their homes in Lower Dir as a result of the military's offensive there.
Al Jazeera sources suggest that number could be at least 50,000.
"Most of these people ... are likely to go to relatives and friends, some of them of course will end up in the camps which the government has set up," Kamal Hyder, Al Jazeera's correspondent in Pakistan, said.
"So the situation, as far as the ordinary people is concerned, is extremely serious."
The government has opened up an emergency camp facility in Jalozai, a former Afghan refugee camp, to receive thousands of people who are fleeing Dir and who may soon be joined by people from Buner.