The blasts left body parts scattered among the graves, witnesses said. About 80 people were injured, including some high-ranking Afghan officials. A spokesman for the Interior Ministry said that the death toll was 15, but it is expected to rise.
"People were standing to pray when the three blasts happened," said one of the mourners, Tahir Gardoon, a professor of political science at Khurshid University. "People were screaming, running, everywhere was fear."
Afghan President Ashraf Ghani condemned the act, calling it "an outrageous attack on mourners burying the martyred. The country is under attack. We must be strong and united."
Saturday's violence capped a week that saw a devastating bomb blast in the heart of Kabul's heavily militarized diplomatic zone that claimed nearly 100 lives and anti-government protests Friday that left six dead. Sporadic gunfire also broke out Saturday near Emergency Hospital in central Kabul.
Continued unrest could paralyze and further weaken Ghani's government, which is facing internal divisions and struggling to contain an aggressive Taliban insurgency and regional militants linked to the Islamic State, experts say.
The Taliban has denied responsibility for both Wednesday's truck bomb blast and Saturday's attack, but the government blames Wednesday's blast on the Haqqani network, seen as the operational wing of the Taliban.
"There is no doubt that this shows once again the total weakness of the security forces," said Hafiz Mansour, a member of the Afghan parliament.
Amrullah Saleh, the state minister for security reforms, survived the funeral attack and, in an interview with the television channel TOLOnews TV, raised questions about the government's ability to secure the besieged capital.
"If there were security measures, then three successive explosions would not have taken place," he said.
The Afghan president earlier had convened an emergency meeting with his security advisers and ordered an investigation into Friday's violent protests to "identify those who acted unlawfully."
On Saturday, protesters held a sit-in at the site outside a hospital near the presidential palace where witnesses said that on Friday palace guards wove through the protesters in armored vehicles, shooting rifles in the air. The protesters are demanding the removal of Ghani's shaky government and pushing for formation of an interim administration, at least partly for its failure to stop the attacks.
Ahmad Zubair Masood, one of the protesters, used the term "Afghan Spring" to describe the demonstrations in Kabul. "This is not Cairo; this is Kabul," he said. "People are arriving from all provinces. Tanks and bullets won't stop them."
While accusing security personnel of using "excessive" force during Friday's protest, some members of the civil society said the rally was hijacked by a group of armed men. The United Nations' special representative for Afghanistan, Tadamichi Yamamoto, has warned that the protests could lead to further violence and strongly urged opportunists not to use the emotional moment to cause instability.
U.S. Embassy Kabul Special Chargé d'Affaires Hugo Llorens suggested in a statement Saturday that "some narrow political elements" have seized the opportunity to "spark violence, resulting in more death and suffering, compounding the grief."
The recent spate of violence comes as President Trump is reviewing the United States' Afghan policy amid a push by some members of his administration to send more troops to Afghanistan to fight the Taliban and other militant groups. About 8,000 U.S. troops are currently deployed in Afghanistan.
Sharif Walid contributed to this report from Kabul.