The fighting broke out after Taliban gunmen with heavy weaponry launched coordinated attacks against NATO’s headquarters next to the US embassy.
The dead included four civilians and two policemen, an Afghan official, who spoke on condition of anonymity, told AFP.
The Afghan government also said at least six Taliban militants were killed, with at least two other militants still resisting after dusk.
Police said gunmen fired rockets and assault rifles from a multi-story building that is under construction near diplomatic missions.
The Taliban militants have claimed responsibility for Tuesday’s attacks.
Meanwhile, the office of Iran’s English-language Press TV news network in Kabul came under attack and one person was injured.
Violence in Afghanistan is on the rise as militants have expedited their spring offensive.
Despite the presence of nearly 150,000 US-led forces in the war-torn country, the violence has grown every passing year since the 2001 US-led invasion.