The coordinated assaults - coming two days after the United States marked the 10th anniversary of the Sept. 11 terror attacks - carried an unsettling message to Western leaders and their Afghan allies about the resilience and reach of the Taliban network.
It was the third major attack in Kabul since late June, casting fresh doubts on the ability of Afghans to secure their own country as the U.S. and other foreign troops prepare to withdraw by the end of 2014.
The American Embassy and NATO both said no staff were wounded. Afghan officials said at least one Afghan police officer, a civilian, and two insurgents had been killed as gunfire and explosions resounded across the city well into the afternoon.
The Interior Ministry said a total of nine people were wounded around the capital. They include four injured by at least two suicide bombings in the western part of the capital.
Four Afghans were wounded when a rocket-propelled grenade hit the original U.S. embassy building next to the new embassy, CIA Director David Petraeus told lawmakers on Capitol Hill. Among them was a small girl who was with a group waiting for visas outside the embassy, he said.
The surge of violence was a stark reminder of the instability that continues to plague Afghanistan nearly a decade after the U.S. invasion that ousted the Taliban for harboring al-Qaida, which carried out the 9/11 plane hijackings.
In Washington, Secretary of State Hillary Rodham Clinton said the U.S. will do everything it can to combat those who committed the "cowardly attack."
Clinton said the U.S. was moving to secure the area and "ensure that those who perpetrated this attack are dealt with." She said the U.S. would assist Afghans injured in the attack.
In Brussels, NATO Secretary-General Anders Fogh Rasmussen said the "enemies of Afghanistan" were trying to disrupt the handing over of security responsibility to the Afghan army and police.
Plumes of smoke rose from the area near the embassy, and U.S. Army helicopters buzzed overhead. The American Embassy is on the edge of the Wazir Akbar Khan area, which is home to a number of other foreign missions. Explosions shook much of the neighborhood.
Meanwhile, gunmen fired from a nine-story office building that is under construction at Abdul Haq square, which is about 300 yards (meters) from the U.S. Embassy. Afghan official said the attack began when about half a dozen insurgents took over the building and began firing toward the embassy and the adjacent NATO headquarters.
Thee military coalition, also known as ISAF, said the insurgents were firing rocket propelled grenades and small arms.
"An Afghan-led response is under way against the attack near the U.S. Embassy and ISAF HQ," NATO said in a statement.
The Kabul police said at least seven insurgents were involved in the attacks around the city. Four were involved in the attack from the building and three attempted to carry out suicide attacks.
All three suicide attackers were killed by police: one on the road leading from the capital to the airport, and two when they tried to attack Afghan police buildings in western Kabul, across the city from the site of the embassy attack. One was shot by police; the bullets detonated his vest and injured two police officers. The other one detonated his vest at a nearby building, wounding two civilians.
"The gunbattle is continuing," Interior Ministry spokesman Sadiq Sadiqi said.
U.S. Embassy spokeswoman Kerri Hannan said that staff had been ordered to take cover in hardened structures. She later issued a statement confirming an attack by gunmen firing rocket-propelled grenades and small arms fire.
"We can confirm there are no casualties at this time among embassy personnel," she said.
NATO also said none of its staff were wounded in the attack. It said the U.S.-led coalition was providing air support to Afghan security forces.
At least one rocket landed on a building housing privately owned Tolo TV and another near a minivan carrying school children.
Associated Press reporters saw police carrying the body of a civilian man, dressed in a white tunic and pants. He was hit by a rocket that landed in the Wazir Akbar Khan neighborhood, police said. A cameraman from Iran's Press TV was wounded by an explosion near their offices in the same neighborhood.
Taliban spokesman Zabiullah Mujahid said a number of suicide bombers attacked Afghan and foreign soldiers at Abdul Haq square. He claimed in a text message sent to reporters that suicide bombers using assault rifles also were attacking the offices of the Afghan intelligence service.
Violence in the once-quiet capital has escalated in recent months.
On Aug. 18 Taliban suicide bombers stormed a British compound in an upscale Kabul neighborhood, killing eight people during an eight-hour firefight as two English language teachers and their bodyguard hid in a locked panic room. Those killed included five policemen, a municipal worker, a security and a New Zealand special forces soldier who was shot in the chest as he tried to free the hostages - who survived.
On June 29, nine insurgents wearing suicide vests stormed the Intercontinental Hotel armed with rifles and rocket launchers on the eve of a major conference on Afghan governance. They killed at least 12 people and held off NATO and Afghan forces for five hours, until U.S.-launched helicopter airstrikes killed the last insurgents hiding on the roof.
Associated Press writers Patrick Quinn and Heidi Vogt contributed to this report.