Witnesses reported three explosions at the burial site of Salim Ezadyar, who was among four people killed on Friday when the protest degenerated into street clashes, with police firing live rounds to disperse hundreds of demonstrators.
The hilly site was littered with bloodied corpses and dismembered limbs, local television footage showed, with one witness telling AFP that "people were blown to pieces" due to the impact of the blasts.
"We don't know what caused the explosions. Initial reports show 15 people have been killed and wounded," interior ministry spokesman Najib Danish said on Twitter.
The funeral was attended by senior Afghan government officials including Chief Executive Abdullah Abdullah, but his office told AFP that he was unhurt.
Kabul city was on lockdown Saturday with armed checkpoints and armoured vehicles patrolling the streets to prevent a repeat of Friday's protests.
Hundreds of demonstrators calling for President Ashraf Ghani to step down clashed with police on Friday, prompting officials to beat them back with live rounds in the air, tear gas and water cannon.
The protesters were voicing anger over a truck bombing on Wednesday in Kabul's diplomatic quarter that killed 90 people and wounded hundreds, the deadliest attack in the city since 2001.
Before the blasts at the funeral, authorities had sealed off roads in the centre of the city, citing the threat of new attacks on large gatherings of people.
"We have intelligence reports that our enemies are trying again to carry out attacks on gatherings and demonstrations," Kabul garrison commander Gul Nabi Ahmadzai said earlier Saturday. "We hope that people will stay away from protests."
But dozens of people still gathered on Saturday under a tent close to the presidential palace calling for Ghani's government to resign, but the assembly was largely peaceful.
"Any government attempt to disrupt our fair and just demonstration will show their complicity with terrorist groups and the perpetrators of Wednesday's attack," said Asif Ashna, a spokesman for the protesters.
"It is the duty of the government to ensure security to the protesters... and the government will be held responsible for any violence."
Kabul has been on edge since the bombing, which highlighted the ability of militants to strike even in the capital's most secure district, home to the presidential palace and foreign embassies that are enveloped in a maze of concrete blast walls.
Residents of the city have demanded answers from the government over the perceived intelligence failure leading to the bombing, which underscores spiralling insecurity in Afghanistan.
Friday's killings will likely further inflame passions as protesters prepared to attend their funeral ceremonies on Saturday.
The United Nations and a host of international allies have urged the protesters for restraint.
"The enemy seeks to manipulate the people's anger and sadness to create division and sow instability," the US embassy said in a statement.
"Now is the time to stand unified and announce to the enemies that Afghans... will not allow cowards to break the resolve to achieve a stable and peaceful nation. The enemies of Afghanistan cannot win. They will not win."