The Islamic State group claimed responsibility for the attack in Jalalabad, the capital of Nangarhar province. Among the four killed were two employees of the NGO, a security guard and an army soldier.
The assault started with a suicide bomber who detonated his explosives' vest at the provincial offices of Save the Children, said Attahullah Khogyani, spokesman for the provincial governor.
Security forces killed four other attackers, he said, adding that at least 26 people, including three members of the Afghan security forces, were wounded.
Two hours later, it was finally over, he said.
"Our primary concern is to secure the safety of all of our staff," Save the Children said in a statement, adding that it had temporarily suspended its work across Afghanistan. In a statement on its Aamaq media arm, the Islamic State group said one of its suicide bombers with an explosive-laden vehicle and a subsequent raid targeted "British and Swedish foundations and Afghan government institutes."
Both the Taliban and IS are active in eastern Nangarhar province.
US State Department spokeswoman Heather Nauert denounced the assault, calling it "heartbreaking" and offered "deepest condolences to the victims and families."
Monica Zanarelli, the International Committee of the Red Cross' head of delegation in Afghanistan, said that an attack against an organisation that helps children is "outrageous."
"Civilians and aid workers must not be targeted," she said. "Increased violence has made operating in Afghanistan difficult for many organisations."
Amnesty International's chief for South Asia, Biraj Patnaik, expressed solidarity with Save the Children following the attack.
"Bombing and shooting people who are working for no other reason than to help improve the lives of young Afghans is a cowardly and despicable act."
The attack followed a deadly weekend siege of the Intercontinental Hotel in the capital Kabul in which 22 people were killed, including 14 foreigners. Multiple US citizens were killed and injured in the Taliban's 13-hour siege of the hotel, the State Department said. No exact figures were immediately available for either the US fatalities or injuries.
Mirwais Samadi, head of the consulate department at Afghanistan's Foreign Ministry, said the Kabul attack was launched by "terrorists" and their supporters.
"Some of our countrymen were martyred and some foreign nationals also were killed," he said. "We express our condolences and thoughts to the victims and families.
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