The Taliban said foreign advisers in the ministry were the targets of the attack, which the al Qaeda-backed Islamist militant group said also involved an exchange of gunfire with ministry guards.
Afghan President Hamid Karzai, who has made repeated peace overtures to the Taliban, said the attack was committed by the "enemies" of Afghans, the presidential palace said.
Karzai, on his way to Turkey, said the attacks showed those responsible wanted to block peace efforts.
Ali Shah Amadzai, deputy police chief for Kabul, said a woman was among those killed.
A doctor at a hospital near the blast site said at least one person was killed and 18 were wounded by the blast, the latest episode in escalating violence in Afghanistan this year which has marked the bloodiest period since the Taliban's ouster in 2001.
The blast damaged part of the first floor of the ministry, which lies several hundred metres away from the presidential palace in central Kabul and forced authorities to evacuate ministry officials.
A police source described the blast as a suicide attack, a rare security breach inside the heavily secured Kabul ministry.
"Yes, it was a suicide attack," the source said, adding two more Taliban attackers escaped from the scene.
The Taliban, overthrown in a U.S.-led invasion in 2001, have stepped up their insurgency despite an increase in the number of foreign troops and have carried out a series of high-profile attacks in Kabul.
The bloodiest suicide attack so far in Kabul came in July, when almost 60 people were killed outside the Indian embassy, including two Indian embassy staff.
Other attacks in Kabul this year included an assassination attempt against President Hamid Karzai in April and the killing of several foreigners in the past two weeks.
Officials say some of the attacks were carried out with the help of members of Afghanistan's security forces.
Thursday's attack comes despite recent proposals for talks with the Taliban.
It echoed growing frustration among many Afghans about insecurity, rampant corruption, lack of the rule of law and civilian casualties caused by foreign troops in strikes against the militants.
The Taliban have ruled out talks until foreign troops, led by the U.S. military and NATO, leave Afghanistan. (Additional Reporting by Jonathan Burch; Editing by Paul Tait)