President Ashraf Ghani Ghani, meanwhile, met with senior security officials during an emergency security meeting.
High-ranking officials, including Acting Foreign Minister Salahuddin Rabbani, were present at the ceremony during the attack.
He also said that "scapegoating Pakistan for failures in Afghanistan will not help" improve the security situation in either country.
The latest killings, which could provoke a new cycle of bloodshed, bring to 101 the number of people killed this week with hundreds injured in one of the worst bouts of violence in the Afghan capital for years.
The blasts occurred where the burial ceremony of Mohammad Salem Izedyar, son of the deputy first speaker of the Senate, was taking place.
There has been no claim of responsibility for the atrocity.
A policeman rests during a demonstration in Kabul, Afghanistan, Friday, June 2, 2017. They had been protesting against Wednesday's massive truck bombing which killed over 100 people and injured more than 600.
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On Friday, demonstrators rushed toward police who fired warning shots and when they attempted to move closer to the palace, police sprayed them with hoses from a water tanker and later fired tear gas.
More than a thousand people demonstrated Friday demanding more security in the capital following a powerful truck bomb attack in the city that killed 90 people and wounded more than 450.
"I urge everyone not to respond to violence with more violence, said Tadamichi Yamamoto, the UN Secretary-General's Special Representative for Afghanistan, expressing heartfelt condolences to family and friends of the victims, all civilians, "who once again suffered indescribable loss today".
"President Ghani condemns the outrageous attack on mourners burying the martyred.The country is under attack.We must be strong and united", he said. There were unconfirmed reports of up to eight people dead and 10 wounded in the melee, which lasted for several hours and shut down central Kabul.
Security authorities in Kabul told people not to attend protests and demonstrations, citing a risk of attacks on large gatherings of people. People had gathered for the funeral one of a protester killed by police on Friday. No group has yet claimed responsibility for the attacks. In a sign of how alarmed Afghanistan's worldwide partners have become at the violence and possible impact on political stability, the United Nations special representative in Afghanistan, Tadamichi Yamamoto, also appealed for restraint.
"The attack was one of the worst suicide attacks to hit the capital since USA and North Atlantic Treaty Organisation forces ended combat operations in the country back in 2014".
The violence and ensuing battles and air strikes caused record civilian casualties in Afghanistan in 2016, when nearly 3,500 civilians were killed and over 7,900 injured.
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