One police officer was killed and five militants were shot dead as security forces hunted down the attackers, with prolonged bursts of gunfire and grenade blasts heard across the Afghan capital.
A United Nations building and several other official premises were caught up in the coordinated assault that started when a suicide car bomb sent a plume of dark smoke into the sky.
Commandos shot dead five gunmen who were firing and throwing grenades from a building, interior ministry spokesman Sediq Sediqqi told AFP.
No other fatalities were immediately reported, but an Italian woman working for IOM was seriously wounded by a grenade explosion.
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"I strongly condemn today's (Friday's) terrorist attack centred on a compound of the International Organization for Migration," Jan Kubis, head of the United Nations Assistance Mission in Afghanistan (UNAMA), said in a statement.
"Three IOM staff members were injured, one seriously. A staff member from the International Labour Organization was also injured."
The Geneva-based IOM, which is affiliated with the UN, works across the world to improve management of cross-border migration.
Yesterday's attack comes a week after another suicide car bomb in Kabul targeted a foreign military convoy, killing 15 people including five Americans.
The Taliban announced their annual "spring offensive" on April 27, opening a crucial period as local security forces take the lead in the fight against the insurgents.
The blasts hit the centre of Kabul at about 4:00 pm local time (1700 IST) yesterday -- Afghanistan's weekly day off -- and fighting continued into the night.
Norwegian soldiers were among the security forces involved in the response operation, an AFP photographer witnessed, but the NATO-led military coalition gave no further details.