Islamic State claimed responsibility for the blast set off by a suicide bomber outside a voter registration center.
Interior Ministry spokesman Najib Danesh said a bomber on foot approached the centre where officials had been issuing identity cards as part of the registration process for voters for the election scheduled for October.
After weeks of relative calm, the blast took place in Dasht-e Barchi, an area of western Kabul inhabited by many members of the mainly Shi'ite Hazara minority, which has been repeatedly hit by attacks claimed by Islamic State.
"There were women, children. Everyone had come to get their identity cards," said Bashir Ahmad who had been nearby the blast, which occurred despite heightened security measures following the January attack.
Voter registration centres have been set up across Afghanistan ahead of long-delayed parliamentary and district council elections due to be held in October and there have been serious concerns that militants might attack them.
President Ghani has been under heavy pressure from his international partners to ensure the elections are held this year, ahead of a presidential election due in 2019 although there has been widespread scepticism that they will take place.
Voter registration began this month but there have already been a number of attacks apparently aimed at disrupting the preparations.
Also on Sunday, a roadside bomb near a voter registration centre in the northern city of Pul-i Khumri killed six members of a family and wounded three as they drove past the site although there was no indication the attacks were linked.
Unless the process of registering millions of voters, many of whom do not have national identity cards, can be completed before winter sets, the vote would almost certainly have to be postponed until next year.
According to UN figures, more than 750 people have been killed or maimed in suicide attacks and bombings by militant groups during the three months to March.