Several blasts and bursts of small-arms fire erupted for about one hour and were ongoing, with the US embassy sounding its "duck and cover" alarm and its loudspeakers warning that the alarm was not a drill.
"It's an ongoing gunfight and blasts, and I can tell you that it's a suicide group attack," Hashmat Stanikzai, Kabul's police chief, told AFP.
Stanikzai said he was unable to confirm the target of the attack, but the explosions came from close to the airport area on the northeast side of the city.
One insurgent detonated himself at compound entrance in the centre of the city at the start of fighting, which left several buildings destroyed or damaged by rocket-propelled grenades, gunfire and explosions.
One policeman, two civilians and all four militants died in that attack, with the government lauding the response of the Kabul security forces for preventing further casualties.
The effectiveness of Afghan security forces is crucial to the government's ability to defeat the Taliban insurgency as NATO-led troops withdraw by the end of 2014.
On Saturday, an Afghan soldier shot dead two US soldiers and one US civilian, the latest "insider attack" to shake efforts by the two armies to work together to defeat the Taliban insurgency.
The killings in the eastern province of Paktika came on the same day that one Italian soldier died when a grenade was thrown into an armoured vehicle in Farah province, in the far west of the country.
