A spokesman for the Taliban, Zabihullah Mujahid, denied in a Twitter post that the group was responsible for the attack.
The Public Health Ministry said another 52 people were wounded in the attack, which was carried out by a bomber on foot. Reports say the number killed may rise.
The Persian new year, known in Afghanistan as Nauruz, is a national holiday, and the country's minority Shiites typically celebrate by visiting shrines. "The area was crowded with civilians and families, a Kabul police officer in the area tells me".
Bloodstains could be seen among scattered belongings on the road at the site of the attack.
TOLOnews journalists at the scene say they saw many bodies lying on the ground soon after the explosion. Sectarian attacks on Shias have increased in recent years.
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One attack, claimed by the Taliban, was the deadliest for months.
This latest suicide attack underscores the growing challenge facing Afghan and foreign forces to protect the already heavily militarised city.
"Kabul is our main effort right now, to harden Kabul, to protect the people of Kabul and the global community that are here because of the strategic impact that has and the importance to the campaign", Nicholson said.
The seemingly endless attacks have undermined support for the government of President Ashraf Ghani, who offered last month to hold peace talks with Taliban insurgents fighting to drive out global forces and reimpose their version of strict Islamic law.
"We had our security in place in and around the shrine", he said.