IS fighters detonated explosive belts amidst a "gathering of Shi'ites" in Deh Mazang, said the group's Amaq News Agency.
President Ashraf Ghani addressed the nation and condemned the attack that also claimed the life of at least one journalist, said Tolo News.
"I will get revenge on those who shed the blood of our citizens," said Ghani.
He ordered Sunday be observed as a day of national mourning.
"Our enemies, by attacking civil liberties, think that they can divide us. But they will fail," he said.
Ghani confirmed that one of three suicide bombers, who had joined demonstrators at the mass rally on Saturday, had been gunned down by security forces before he could detonate his explosives.
"I have directed a special commission and the attorney general to investigate the incident and find those responsible," he said.
The incident took place as thousands of demonstrators gathered for a sit-in protest at Deh Mazang against the route of a planned power line project.
The Taliban militant group has denied any hand in the deadly attack. Taliban group spokesman Zabiullah Mujahid said the group has no link with the explosions in Kabul city, calling it a move to create divisions among the people, said Khaama.com.
The Ministry of Interior on Saturday night confirmed the death toll had risen to 80 and that over 230 people were wounded.
The United Nations Assistance Mission in Afghanistan (UNAMA) condemned the attack.
In a statement issued by the organization, it said the two explosions occurred at Dehmazang Circle in Kabul city "targeted a peaceful demonstration, causing horrendous levels of civilian harm".
"This attack is particularly heinous because it targeted civilians as they exercised their rights to freedom of assembly and freedom of expression," the statement read.
According to Tolo News, police said one suicide bomber detonated his explosives while the second person failed to explode his suicide vest. The third bomber was killed by security forces at the scene before he could detonate his vest.
Photographs and video footage taken by witnesses immediately after the blasts painted a horrific scene.
Hundreds of people, many covered in blood, were filmed running from the scene. Extensive damage was also caused to nearby buildings, said Tolo News.
The demonstrators had gathered in a protest organised by the Enlightening Movement over the planned route of the 500kV power line project that the Afghanistan government intends to roll out.
The project will see a new power line run from Turkmenistan through Salang to Kabul. The Enlightenment Movement wants the line to be routed through Bamiyan to Kabul, Tolo News reported.
Amnesty International has also condemned the attack.
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