Updated

WASHINGTON -- The Obama administration declared Tuesday it wouldn't allow the brazen assault on the U.S. Embassy and other buildings in Kabul to deter the American mission in Afghanistan, warning the attackers that they would be relentlessly pursued.

The White House said President Obama was briefed on the attack Tuesday morning, after insurgents fired grenades and assault rifles at the embassy, NATO headquarters and other sites in Afghanistan's capital, while suicide bombers struck police buildings. The Taliban claimed responsibility.

Secretary of State Hillary Rodham Clinton called it a "cowardly attack."

"We will take all necessary steps, not only to ensure the safety of our people, but to secure the area and to ensure that those who perpetrated this attack are dealt with," Clinton told reporters.

The coordinated assaults come two days after the U.S. marked the 10th anniversary of 9/11 and carried an unsettling message to Western leaders about the Taliban's resilience. It was the third major attack in Kabul since late June, casting doubts on U.S. plans to stem violence in the country and withdraw all troops by 2015.

CIA Director David Petraeus said all embassy personnel were accounted for and no one was injured.

The attack was launched by a "handful of attackers, five or so, apparently armed with suicide vests" and rocket-propelled grenades, according to Petraeus, who said he spoke to the U.S. Ambassador to Afghanistan, Ryan Crocker, earlier Tuesday.

Speaking before a combined session of congressional intelligence committees, Petraeus said that four Afghans were injured when a rocket-propelled grenade hit the main building in the embassy complex. The injured included a small girl who was waiting outside for a visa. The girl was treated at the military's hospital nearby, he said.

Speaking at the State Department, Clinton paid tribute to the civilians who work alongside State Department diplomats and USAID development experts in Kabul, assisting Afghans "in a transition toward stability, security and prosperity." She called them "dedicated brave men and women."

"They will not be intimidated by this kind of cowardly attack," Clinton said. "They work hard every day along with their Afghan colleagues to help children go to school, to help save mothers' lives at childbirth, to build roads, to assist farmers."

Still, she acknowledged the challenge of combating the Taliban and its insurgent allies who are "engaged in a constant effort to threaten and undermine peace and progress."

"We will be continuing with even greater commitment to doing all we can to give the Afghan people, who suffered so much, a chance at a better future for themselves and their children," Clinton said.