The Taliban claimed responsibility for the attack near the Afghan intelligence agency office, which sent clouds of acrid smoke billowing in the sky and rattled windows several miles away.
It was not immediately known if there were any casualties from the assault, which marks the first major Taliban attack in Kabul since the insurgents launched this year's fighting season.
"The first blast was carried out by a suicide bomber in a car and possibly one or two bombers are still resisting," interior ministry spokesman Sediq Sediqqi told AFP.
Taliban spokesman Zabihullah Mujahid claimed their fighters had managed to enter the office of the National Directorate of Security, the main Afghan spy agency.
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Afghan officials did not confirm that claim but intense gun battles could be heard near the NDS compound. The Taliban are generally known to exaggerate battlefield claims.
"Today's terrorist attack near Puli Mahmood Khan area of Kabul city demonstrates the clear defeat of the enemy in the face-to-face fight against Afghan security forces," the Afghan presidential palace said in a statement on Twitter.
The Taliban warned they would "employ large-scale attacks on enemy positions across the country" during the offensive dubbed Operation Omari in honour of the movement's late founder Mullah Omar, whose death was announced last year.
The insurgents began the fighting season last week by targeting the northern city of Kunduz, which they briefly captured last year in a stunning setback for Afghan forces.
But officials said Afghan security forces drove Taliban fighters back from the city on Friday.
The Taliban's resurgence has raised serious questions about Afghan forces' capacity to hold their own, with an estimated 5,500 troops killed last year, the worst ever toll.
Peace talks which began last summer were abruptly halted after it was revealed that Taliban leader Mullah Omar had been dead for two years.
A four-country group comprising Afghanistan, the United States, China and Pakistan has been holding meetings since January aimed at jump-starting negotiations, though their efforts have so far been in vain.