A spokesperson told the BBC that 10 of the dead were civilians, and that eight people were injured.
The Taliban has said it carried out the attack, which happened as offices closed for the day.
Earlier, an attack near a market in the eastern Afghan province of Kunar killed at least 11 people, officials said.
It remains unclear exactly how the Kabul attack unfolded, although officials said the attacker was on foot.
In Kunar, the provincial governor's office said the bomber rode up on a motorcycle to the entrance of a government office in the provincial capital, Asadabad, before detonating his device.
Some of the victims were reported to be children playing in a nearby park.
Reuters news agency reported that a tribal elder and militia commander named Haji Khan Jan was among the dead.
He had been closely involved in a number of operations against the Taliban last year.
In recent months, there has been an upsurge in fighting between the Taliban and government forces.
President Ashraf Ghani was quoted by local media as saying he will not hold peace talks with groups that kill civilians.
A suicide bombing near the defence ministry in the Afghan capital of Kabul has killed at least 12 people, the government says.