Afghan officials say the attack killed 29 people and wounded 52, updating an earlier toll.
In an interview from Kabul, Habib Wardak, a national security analyst, said the timing and location of the attack was "no coincidence". IS said the attack targeted "a gathering of Shiites celebrating Nowruz".
The attack happened as many Afghans are marking the Nowruz holiday, a celebration of the start of spring and the Persian new year. The country's Shiite minority typically marks the holiday by visiting such shrines.
The blast occurred outside Ali Abad hospital, close to Kabul University in the west of the city, he said.
Daud said the attacker managed to slip past police checkpoints set up along the road.
The Islamic State claimed responsibility for this attack via its Amaq website, though both the Taliban and ISIS have carried out attacks in the capital in recent months, spreading terror and fear across Kabul.
"We had our security in place in and around the shrine", he said.
The bomber reportedly walked toward the Sakhi shrine and blew up his explosives when identified by the police.
Afghanistan's Shiite population has repeatedly been targeted by IS affiliates, who view the minority group as apostates of Islam. Several women and children were among the casualties.
During midday, a suicide bomber struck on the road and blew himself near a Shiite shrine in Kabul, on March 20.
On Saturday, a Taliban suicide bomber detonated explosives near a compound for foreign security contractors in Kabul, killing three civilians. Officials reveal that 26 people were killed and 18 others were injured in the terror attack.
The Taliban has been resurgent since the withdrawal of US-led North Atlantic Treaty Organisation combat troops at the end of 2014, taking back territory and devastating Afghanistan's beleaguered security forces.
Afghan President Ashraf Ghani condemned the attack in a statement, calling it a "crime against humanity".