Security forces are fighting back, he said, adding that six civilians have been killed.
On May 6, an explosion at a mosque in the eastern Afghan province of Khost killed at least 10 people and wounded 29.
It was the latest in a series of high-profile attacks that have killed and wounded hundreds of civilians in Afghanistan this year and put heavy pressure on the Western-backed government of President Ashraf Ghani.
Earlier, he said there had been two explosions.
In Sunday's attack, one militant blew himself up at the gate of the building in the centre of the city, allowing the others to enter, said Sohrab Qaderi, a member of the provincial council.
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There was no immediate claim of responsibility for the attack.
A vehicle bomb was detonated at the entrance to the state accounts office before a group of about six attackers armed with machine guns and rocket-propelled grenades rushed the building, the officials said.
Jalalabad is the capital of restive Nangarhar, which borders Pakistan.
Taliban militants and fighters of Islamic State outfit have presence in Nangarhar province, some 120 km east of Kabul.
Nine journalists, including AFP's chief photographer in Kabul Shah Marai, were among 25 people killed in a double suicide blast claimed by IS in the capital on April 30.