It began around 12.50 pm (0820 GMT) when the militants detonated a auto bomb at the entrance to the city finance department, "opening the way for a number of armed attackers to enter the building", Nangarhar provincial governor's spokesman Attaullah Khogyani told AFP.
Governor Spokesman Attaullah Khogyani told Pajhwok Afghan News nine people have been killed and 39 others injured so far in the attack.
Public health officials said that at least 15 people, including a child, had been killed and 42 wounded.
All four assailants were killed by security forces after five-hour gun battle.
Earlier, he said the attack had been carried out by four attackers and that there had been two explosions. Then more explosions were then heard coming from inside the building, witnesses said.
Spain nearly banned from this weekend's Eurovision Song Contest — SNAP JUDGEMENT
The right to host the contest came after Salvador Sobral won the 2017 edition in Kyiv with his song Amar Pelos Dois . It comes after Jessica told Stellar magazine that she still gets nervous before taking to the stage.
Jalalabad is the capital of restive Nangarhar, which borders Pakistan.
The militant attack is the latest in a series across Afghanistan this year. He said security forces had arrived at the site engaging the insurgents.
Violence has escalated across Afghanistan since the announcement of the Taliban's annual spring offensive last month and there have been heightened security fears around preparations for elections in October.