The missiles fell in northern Iraq's Erbil, which hosts US-led coalition troops, officials say, but no casualties were reported.
Iran's Revolutionary Guard has said that they targeted an Israeli "strategic centre" in Iraq with missiles, after Kurdish Regional Government (KRG) in northern Iraq reported cross-border missile fire.
A "strategic centre for conspiracy and mischiefs of the Zionists was targeted by powerful precision missiles fired by the Islamic Revolutionary Guards Corps.", it said in a statement on Sunday.
Earlier, security forces in northern Iraq said "a dozen ballistic missiles" targeted the city of Erbil, including US facilities, causing damage but no major casualties in the early hours of Sunday.
Sunday's attack on Erbil comes nearly a week after two officers from Iran's Revolutionary Guards were killed in Syria in a strike attributed to key US ally Israel.
The Guard had already warned last week that Israel, Iran's arch enemy, "will pay for this crime".
The Guard's statement on Sunday said: "once again, we warn the criminal Zionist regime that the repetition of any mischief will face harsh, decisive and destructive responses."
American Consulate in Erbil, Iraq, under missile attack. pic.twitter.com/w9BFN8XUZX— Sebastian Gorka DrG (@SebGorka) March 13, 2022
Two rockets targeted Baghdad's ultra-secure Green Zone that houses the US embassy early on Sunday, Iraq's security forces said in a statement.
"The Green Zone in Baghdad was the target of two Katyusha rockets. The first was shot down in the air by C-RAM defence batteries, the second fell in a square, damaging two vehicles," the statement said.
A security source said that the shot down rocket fell near the US embassy, while the second came down roughly 500 metres (1640 ft) away.
The US State Department reported that there were no casualties or injuries in the attack.
In recent months, dozens of rocket assaults or drone bomb attacks have targeted US troops and interests in Iraq.
The attacks are rarely claimed but are routinely pinned on pro-Iran factions in Iraq.