Many in the crowd were minority Shia.
One of the city's largest shrines, it is a focus of New Year prayers in Kabul every year.
Nasrat Rahimi, the deputy spokesman for the Interior Ministry, said another 18 people were wounded in the attack.
Taliban and Islamic State militants have increasingly targeted the war-weary city in recent months as USA and Afghan forces ramp up air strikes and ground offensives against the groups.
Kabul had been on alert for attacks over the Nawruz holiday but the bomber was still able to detonate his explosives as people were leaving the Kart-e Sakhi shrine, in a heavily Shi'ite area in the west of the city.
Russian diplomats prepare to leave United Kingdom after spy poisoning
Last week, Britain, France, Germany and the United States issued a joint statement blaming Russian Federation for the attack. The 28 leaders meeting in Brussels on Thursday will base their response on what answers Moscow provides, the draft says.
There has been no immediate claim of responsibility, but the Hazara community has been on the hit list of pro-Daesh militants.
Some Western and Afghan security experts believe the Haqqani Network, a hard-line wing of the Taliban, has been behind some of the attacks in Kabul in recent months, including those claimed by IS.
United States and Afghan officials say this is in response to air strikes and other security operations which have pushed the militants back in some areas.
Afghan officials say the attack killed 29 people and wounded 52, updating an earlier toll.
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