The IS-linked Amaq news agency said two fighters "detonated explosive belts at a gathering of Shias" in Kabul.
The attack in Deh Mazang Square targeted thousands from the Shia Hazara minority who were protesting over a new power line, saying its route bypasses provinces where many of them live, BBC reported. The Taliban have condemned the attack. Spokesperson Zabiullah Mujaheed sent an email to the media saying they were not behind it.
A freelance journalist working for BBC Afghan said blood and body parts were everywhere, with debris strewn around.
Afghan President Ashraf Ghani said he was "deeply saddened", adding: "Peaceful protest is the right of every citizen, but opportunist terrorists infiltrated the crowds and carried out the attack, killing and injuring a number of citizens, including some security forces."
A large part of Kabul's city center had been sealed off for the protest march.
According to Financial Tribune, the demonstrators had waved banners and chanted "death to discrimination", angry that the 500 kV power transmission line from Turkmenistan to Kabul would not pass through Bamyan and Wardak provinces, which have large Hazara populations.
The Hazaras, mostly Shia Muslims, make up the third largest group in Afghanistan. They live mainly in the center of the country.
They complain of persistent discrimination, especially during Taliban rule in the late 1990s, when many of them fled to Pakistan, Iran and Tajikistan.
EghtesadOnline: The self-styled Islamic State terrorist group has said it was behind an attack on a protest march in the Afghan capital, Kabul, that killed at least 61 people and wounded 200.