Attahullah Khogyani, the spokesman for the provincial governor, said the attackers killed at least four victims and that at least 20 other people were wounded.
At least four attackers armed with rocket-propelled grenades and machine guns were fighting police, he said.
Following the explosion, several gunmen opened fire and engaged in clashes with the security forces near the government facility.
Last year, United States stepped up its support to struggling Afghan forces, announcing plans for thousands of additional advisers and increasing air strikes in a bid to force the Taliban to enter peace negotiations.
Earlier, an explosion at a mosque in the eastern Afghan province of Khost killed at least 13 people and wounded 31 and the casualty total may rise.
Khogyani further added that the attack was carried out a by group of eight militants with at least two of them detonating their explosives near the Customs Department at around 12:30 pm local time.
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Jalalabad is the capital of restive Nangarhar, which borders Pakistan.
Daesh has claimed responsibility for an attack on a government building in Afghanistan on Sunday, the militant group's Amaq news agency said without providing any evidence. Some areas of the province are a stronghold of the Islamic State group, while Taliban fighters are also active there.
Dozens of people have been killed in voter registration centres in recent weeks.
The Taliban recently launched their annual spring offensive, in an apparent rejection of a peace overture by the Afghan government.
Heavy security forces arrived at the spot of the attack.