Initial reports suggest the bomber attacked a crowd of hundreds of people who had gathered to celebrate the start of Nowruz, the New Year festival.
Many in the crowd were minority Shia. Reports say the number killed may rise.
The Islamic State (IS) group claimed the bombing. Sectarian attacks on Shias have increased in recent years.
Many have been attributed to IS.
An interior ministry spokesman told the BBC a suicide attacker approached the Sakhi shrine on foot and detonated his explosives when identified by police.
The blast occurred outside Ali Abad hospital, close to Kabul University in the west of the city, he said.
Eyewitnesses told BBC Afghan that many young people who were dancing, singing and celebrating were among the casualties.
The shrine has been attacked before. At least 14 people were killed in October 2016 as Shia Muslims prepared for a religious day of mourning during the festival of Ashura. A bombing in 2011 killed at least 59.
One of the city’s largest shrines, Sakhi is revered by many Afghan communities but especially by the Shia minority. It is a focus of New Year prayers in Kabul every year. (Courtesy BBC)