The Kabul bombing occured in the Dasht-e-Barchi area, where numerous country's Shia Hazara minority reside.
Baghlan officials also confirmed at least five people were wounded in the explosion.
This latest suicide attack in Kabul underscores the growing challenge facing Afghan and foreign forces to protect the already heavily militarized city.
"I wish Allah's mercy upon those who martyred, speedy recovery to the wounded, and convey my deep condolences to victims' families". It stands alongside the Afghan people in the fight against terrorism.
Meanwhile Antonio Guterres, the UN Secretary-General, while condemning the attack said, "They must not be allowed to succeed in deterring Afghan citizens from carrying out their constitutional right to take part in forthcoming elections".
On behalf of the USA government, I strongly condemn today's suicide attack on a voter registration center in Kabul that killed approximately 60 people and wounded over 100 others.
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A stun gun had little effect on him, possibly because it did not attach properly through his clothing, the arrest report said. Chicago's Jacob Garcia, 28, first received special attention after a female passenger accused him if improperly touching her.
All victims were members of the same family, Tolo News reported. Last week three other attacks took place.
They (the government) should be keeping the country safe.
Voter registration centres have been set up across Afghanistan ahead of the long-delayed parliamentary and district council elections due to be held in October, and there have been serious concerns that militants might attack them.
Another witness to the attack, named Akbar, told Tolo TV: "Now we know the government can not provide us security". The blast occurred in front of a local school where a voter registration site is located.
President Ghani - under global pressure to hold long-delayed parliamentary elections this year - condemned the attack and said it "cannot divert us from our aims or weaken this national democratic process".
The groups want to establish a harsh form of Islamic rule in Afghanistan, and are opposed to democratic elections.