The attack is an example of increasingly frequent "complex assaults" in which militants create chaos with an initial blast, then storm a sensitive building, which leads to an hours-long gun battle, extending the bloodshed.
Fifty-one children in two schools near the blast site were hurt by flying glass, said Education Ministry spokeswoman Nooria Nazhat.
The attack comes as USA special peace envoy for Afghanistan Zalmay Khalilzad holds a seventh round of peace talks with the Taliban Islamist militant group in Qatar, aimed at bringing an end to the 18-year war in Afghanistan.
The ministry did say 100 people were wounded, half of them children.
During a visit last week to the Afghan capital, U.S. Secretary of State Mike Pompeo said Washington would like to see an agreement before September 1, considered an ambitious deadline by analysts but likely linked to Afghan presidential polls scheduled for later that month.
"Clashes ended with the death of all five attackers", said interior ministry spokesman Nasrat Rahimi, adding that more than 210 people had been rescued during the operation.
At least three gunmen entered a building around the defence ministry after the explosives were detonated near the ministry's engineering and logistics department, a government security official said.
Abdullah said the country's "heroic" security forces will continue to "pursue & punish" Taliban militants, whom he called "miscreants".More news: Sanders joins with House Democrats on bill to cancel student debt
More news: Royal split: Prince William, Harry drift further apart in charity move
More news: Darius Garland A 'Strong Consideration' To Go No
AFP reporters could hear gunshots and multiple smaller explosions as fighting continued almost seven hours after the first blast.
"We were sitting inside the office when the world turned upside down on us", said Zaher Usman, an employee at a branch of the culture ministry, which he said stands just 150 metres (yards) from the blast.
Afghans carry a dead body after a vehicle bomb blast near a governmental institution in downtown Kabul, Afghanistan, July 1, 2019.
'When I opened my eyes, the office was filled with smoke and dust and everything was broken, my colleagues were screaming, ' Usman said. A media watchdog said seven Shamshad journalists were among the wounded.
Both the Taliban and the Islamic State (IS) terrorist group operate in Kabul, which has been relatively quiet in recent months following a spate of violent explosions, many claimed by the IS affiliate that operates in Afghanistan. There was no immediate confirmation if they would go ahead after the blast.
Suhail Shaheen, spokesman for the Taliban's political office in Doha, told The Associated Press on Monday that "our main concern is to make sure a timeline for troop pullout is announced".
The explosion came as the U.S. was set to begin a third day of negotiations with the militants in Doha.
A potential deal would see the United States agree to withdraw its troops after more than 17 years in Afghanistan, igniting deep concerns among Afghans who fear Washington will rush for the exits and allow the militants to return to some semblance of power.