Attaullah Khogyani, a provincial government spokesman, said security forces stormed the compound.
A police officer was killed in the attack along with three Save the Children workers. One insurgent was also killed by police while another was still fighting, Khogyani said.
Nearly 46 people in the compound, many of whom hid in a safe room in the building, were rescued.
In an official statement from Save the Children, the organization confirmed "three staff members were killed" in their office attack.
"We have temporarily suspended our operations across the country following today's events, however we remain fully committed to helping the most deprived children of Afghanistan."
The organization said they were "shocked and appalled at the violence, carried out against our staff in Afghanistan who are dedicated humanitarians, committed to improving the lives and wellbeing of millions of children across the country."
The Swedish Ambassador to Afghanistan, Tobias Thyberg, condemned the attack on Twitter.
"Attacking those who risk their lives to bring development and humanitarian aid to Afghanistan is an unacceptable crime against international law and an affront to the people of Afghanistan and their friends," Thyberg said. "Our thoughts go out to #Jalalabad."
The Taliban distanced themselves from the attack, but the Islamic State claimed responsibility Wednesday.
The Save the Children Fund -- an international non-governmental organization that promotes children's rights -- provides relief and support to children in developing countries.
The charity organization has been working in Afghanistan since 1976 and helps more than 700,000 children in the country.