The bombing, the city's deadliest so far this year, took place on Dar-ul Aman road, where the streets are currently illuminated with colourful lights at night for the upcoming 100th anniversary of Afghanistan's independence.
Speaking through the Nashir News Agency, an online forum linked to Islamic State, the group said a meeting of "heretics" was targeted and 400 were left dead and injured. 
Islamic State considers Muslim Shiites to be heretics and has targeted them since its birth in Afghanistan in early 2015.
The United Nations Assistance Mission in Afghanistan (UNAMA) condemned the bombing and described it as a deliberate attack against the Shiite community. 
"An attack deliberately targeting civilians is an outrage, and deeply troubling, as it can only be described as a cowardly act of terror," the UN Secretary General's Special Representative for Afghanistan, Tadamichi Yamamoto, said in a statement.
Sediq Seddiqi, a spokesman for Afghan President Ashraf Ghani, wrote on Twitter that he was "devastated by the news of a suicide attack," while Afghan Chief Executive Abdullah Abdullah "strongly" condemned the "heinous and inhumane" attack.
ToloNews, a local TV news channel, posted a photograph of relatives digging graves for 14 members of one family who were killed in the bombing.  
Jawad Jalali, a photographer with Italian NGO Emergency, which operates a hospital in Kabul, cited a crying mother saying her husband and two of her sons were at the wedding.
"Oh god my son, I beat my son with a wire today because he was being naughty, oh god I wish I didn't!!!", the mother was quoted by Jalali as saying.
According to Jalali, the woman lost consciousness after the hospital read out a list of those killed in the attack. 
The attack was the 17th to take place in Kabul since the beginning of the year and the second deadly one in August alone. At least 176 people have died and at least 914 others have been injured in the attacks.
Meanwhile, between 10 and 12 civilians were also killed after a vehicle struck a roadside bomb in Afghanistan's northern Balkh province on Sunday morning, police spokesman Adil Shah Adil told dpa. 
The incident took place in Dawlat Abad district near the border with Turkmenistan around 6 am (0130 GMT) as the car was driving from the district towards Mazar-e Sharif city, the capital of the province, Adil said. 
Roadside bombs are often laid by militants to hamper the movement and progress of Afghan forces, but they frequently harm civilians.
Non-suicide improvised explosive devices (IEDs) accounted for 20 per cent of all civilian casualties in the first six months of the year, according to a report by the United Nations Assistance Mission in Afghanistan published at the end of July. 
In total, nearly 1,400 people were killed and 2,500 others injured between January 1 and June 30, according to the report. 
Dozens of Afghan security forces and Taliban militants also die in combat every day as talks about peace negotiations continue, while more than 220,000 civilians have had to flee their homes due to the ongoing fighting since the beginning of the year. 
The US and the Taliban have been meeting since the middle of last year in a bid to find a peaceful solution to the Afghan conflict, which has claimed the lives of tens of thousands of Afghans since the US first invaded in 2001.
Parties to the conflict believe that their own bargaining position can be strengthened by increased military pressure.
Many Afghans, however, fear that even after an agreement is reached the violence might not abate, since the Islamic State is not a part of the negotiations and will remain a threat to the security of the country. 
Afghans fear that a number of disgruntled Taliban members may also join the terrorist group to keep fighting.
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