In recent months terrorists claiming allegiance to Daesh had largely appeared to be bottled up in a mountainous area along the border with Pakistan under threat of US airstrikes.
The latest attacks indicate the group remains a potent threat to a government already battling an insurgency dominated by the rival Taliban.
At least a dozen Afghan security forces and civilians had been killed, with another 18 wounded, Nangarhar province governor Saleem Khan Kunduzi said in a statement reported by Reuters.
Local officials claimed more than 100 Daesh terrorists had been killed in fighting in Nangarhar over the past three days, although exact figures varied and could not be independently verified.
"There is no doubt that Daesh do not respect anyone," Kunduzi said. "They kill people, regardless of whether they're a child or a woman. They burn down madrasas, mosques and schools."
As many as 25 homes had been burned down in Kowt district, and five civilians were reported kidnapped, Nangarhar officials said.
Hundreds of police and soldiers are engaged in the area with reinforcements on the way, provincial police chief Zarawar Zahid said.
At a small event on Sunday in Kabul, the head of Afghanistan's civil society federation, Sediq Ansari, blamed local leaders for being negligent in the face of Daesh threats and called on them to be suspended.
"They should be accountable for every drop of blood that has been shed in Nangarhar so it becomes a lesson to other officials," he told gathered reporters.
Militants linked to Daesh have not made as much progress in Afghanistan as in Syria and Iraq, where the group seized major cities and wide swaths of territory and attracted thousands of recruits.
In Afghanistan, the group is thought to consist mostly of disaffected members of other insurgent movements, including the Taliban, who have often battled Daesh for control of areas in Nangarhar.