US forces in Iraq and Syria were also targeted with rockets and drones at least five more times on Friday - three times at separate bases in Syria, and twice at the Ain al-Asad airbase west of Baghdad, a different US defence official said.
The attacks were the most recorded against US forces in the region in a single day since mid-October, when Iran-aligned militias started targeting US assets in Iraq and Syria over Washington's backing of Israel in its war against Hamas in Gaza.
US Defense Secretary Lloyd Austin, in a call with Iraq's Prime Minister Mohammed Shia Al-Sudani, condemned the attacks and singled out Iran-aligned armed groups Kataib Hezbollah and Harakat Hezbollah al Nujaba for the recent targetting of US personnel.
"The United States reserves the right to respond decisively against those groups," Austin told Sudani, according to a Pentagon statement summarising the call.
The embassy attack early on Friday marked the first time it had been fired on in more than a year, apparently widening the range of targets. Dozens of military bases housing US forces have been attacked, increasing fears of a broadening regional conflict.
No group claimed responsibility, but previous attacks against US forces have been carried out by Iran-aligned militias operating under the banner of the Islamic Resistance in Iraq.
The US military official, speaking on the condition of anonymity, left open the possibility that more projectiles were fired at the embassy compound but did not land within it.
The US officials said Friday's attacks caused no injuries, and the embassy attack caused very minor damage.
Explosions were heard near the embassy, in the centre of Baghdad, at about 4am on Friday. Sirens calling on people to take cover were activated.
State media said the attack damaged the headquarters of an Iraqi security agency.
Sheikh Ali Damoush, a senior official in the Lebanese group Hezbollah, said in a Friday sermon that attacks by Iran-aligned groups across the Middle East aim to apply pressure for a halt to Israel's offensive in the Gaza Strip. He did not refer specifically to Friday's attack.
US forces in Iraq and Syria have been attacked at least 84 times since October 17, the defence official said.
The US has responded with a series of strikes that have killed at least 15 militants in Iraq and up to seven in Syria.
The State Department called on the Iraqi security forces to immediately investigate and arrest the perpetrators.
"The many Iran-aligned militias that operate freely in Iraq threaten the security and stability of Iraq, our personnel, and our partners in the region," State Department spokesperson Matthew Miller said in a statement.
Sudani directed security agencies to pursue the perpetrators, describing them as "unruly, lawless groups that do not in any way represent the will of the Iraqi people", a statement from his office said.
He also said that undermining Iraq's stability, reputation and targeting places Iraq has committed to protect were acts of terrorism.
The head of militia Kataib Sayyid al-Shuhada, one of the main factions that has been targeting US forces in the region, said in a social media post that he rejected "stopping or easing operations" while "Zionist crimes continue in Gaza".
Aside from its diplomatic staff in Iraq, the US has about 2500 troops in the country on a mission it says aims to advise and assist local forces trying to prevent a resurgence of Islamic State, which in 2014 seized large swathes of both Iraq and Syria before being defeated.
Iran-aligned Houthis in Yemen have been firing at Israel and ships in the Red Sea in a campaign they say aims to support the Palestinians. US warships have shot down several of their projectiles.
