"No security force personnel or civilian was harmed in the attack," Salangi said.
The militants used rocket-propelled grenades (RPGs) and heavy weapons to enter the eastern gate of the palace, the police chief said.
An explosives-packed car was also detonated by the attackers, witnesses said.
They confirmed that the militants, using false entry cards, had passed the first gate.
Taliban claimed responsibility for the attack, which they said inflicted casualties on Afghan and US forces.
Explosions and firing occurred at the presidential palace around 6.30 a.m. local time (about 7.30 a.m. Indian Standard Time), and heavy gunfight ensued.
Local TV channels reported at least 12 explosions in the Shash Darak area, the eastern edge of the presidential palace.
The explosions came hours ahead of a meeting Afghan President Hamid Karzai was to have with journalists, and many press people were at the site at the time of the attack.
The CIA station and the Afghan ministry of defence are in the vicinity of the presidential palace.
Government troops cordoned off the area immediately after the blast, and smoke was seen rising from the site. It was not clear if the president was in the palace at the time of the attack.
The incident came as the US Special Envoy for Afghanistan and Pakistan James Dobbins is visiting Kabul for talks with the Afghan leaders over the stalled peace talks with the Taliban.
The Afghan government, angry that the Taliban office in Doha, Qatar, appeared more like a quasi-embassy, with the flag of the Taliban atop it, suspended negotiations with the US on an accord on US military presence in Afghanistan after 2014.
Taliban opened their political office in Doha June 18 but the first round of peace talks between Taliban and the US representatives was cancelled the same day.