The attack wounded more than 80 people, many of whom suffered severe burns.
US Ambassador to Afghanistan John R. Bass said: "The United States continues to stand with all Afghans in their pursuit of peace, security, and prosperity".
The Afghanistan branch, which calls itself the Khorasan Province, has stepped up attacks over the past year even as the extremists' self-styled caliphate in Syria and Iraq has collapsed.
1TV News, quoting an interior ministry spokesman, said a suicide bomber was likely to have been behind the attack.
They took place during a ceremony to mark the 38th anniversary of the Soviet invasion in the country.
Health Ministry spokesman Ismail Kawasi confirmed the toll.
Islamic State has expanded its presence in Afghanistan since it first appeared in the region in 2015.
The center is located in the Shiite-dominated Dasht-e-Barchi neighborhood in the west of the capital.
According to them, there were at least two, maybe three suicide bombers who also used grenades in the attack.
The bomb reportedly killed dozens of students after exploding near the offices of a news agency. It also hit the Afghan News Agency, which has an office on the floor above the center.More news: Trump Is America's 2nd Most-Admired Man
Aamaq in its statement also claimed the center was funded by Iran and propagating Shiite beliefs.
Outraged by the killings, Ghani denounced the suicide assault as a crime against humanity, an attack on Islam and all human values.
Interior Ministry deputy spokesman Nasrat Rahimi said at least 40 people had been killed and 30 wounded in the blast, with numerous victims being students.
"The target of the attack was the Tabayan cultural centre". Two more blasts followed at the same building, but caused no injuries.
No one immediately claimed responsibility for the attack but the local Islamic State affiliate has carried out several attacks targeting Shiites in Afghanistan.
According to officials, women, children and journalists are among the dead.
Dawlat Abad District Gov. Mohammad Karim said Thursday the powerful mine killed six shepherd children Wednesday ages 10, 9 and 8.
"This gruesome attack underscores the dangers faced by Afghan civilians", Reuters reported rights group Amnesty International saying in a statement from its South Asia Director, Biraj Patnaik.
Afghanistan has the highest number of mine victims in the world, which along with other roadside bombs kill or wound an estimated 140 people every month.