The missiles targeted the Ain al-Assad base in Anbar province and a facility near Erbil's airport in northern Iraq early on Wednesday morning; they were fired in retaliation for the killing of top Iranian commander Qassem Soleimani by the US, Iran said.
US President Donald Trump said he would make a statement on the attacks on Wednesday morning in Washington, DC.
As tension increases, governments around the world are calling for a return to diplomacy and considering plans to withdraw their citizens.
The assassination of Soleimani on January 3 was a major escalation in already deteriorating relations between Iran and the US.
The general - who controlled Iran's proxy forces across the Middle East - was regarded as a terrorist by the US government, which says he was responsible for the deaths of hundreds of American troops and was plotting "imminent" attacks.
Iran vowed "severe revenge" for his death.
Mr Trump, meanwhile, warned the US would respond in the event of retaliation "perhaps in a disproportionate manner".
"He was a monster. And he's no longer a monster. He's dead," Mr Trump said, defending his decision. "He was planning a big attack, a bad attack for us. I don't think anyone can complain about it."
Millions of Iranians turned out for the commander's funeral, with mourners chanting "death to America" and "death to Trump".
A stampede at the burial in Soleimani's hometown Kerman killed 50 people and injured 200 more on Tuesday.
After the event resumed, top Iranian officials renewed their threats of revenge.
"The martyr Qasem Soleimani is more powerful... now that he is dead," the Revolutionary Guards' top general, Maj Gen Hossein Salami, told the crowds.
How does Iraq fit into this?
Iran supports a variety of Shia militia groups in neighbouring Iraq. On Friday, Soleimani had just arrived at Baghdad airport and was travelling in a convoy alongside officials from such groups when their cars were hit by several US missiles.
Iraq now finds itself in a difficult position as an ally both of Iran and of the US. Thousands of US troops remain in the country to assist in the broader struggle against the Sunni Muslim Islamic State (IS) group but Iraq's government insists the US has acted beyond the terms of this agreement.
Prime Minister Adel Abdul Mahdi labelled the missile strike that killed Soleimani as a "brazen violation of Iraq's sovereignty and a blatant attack on the nation's dignity".