"From now on, any new evil act by the USA will face a stronger and more crushing response with a broader range", he added.
The sources said as many as 20 critical points in the base were hit by 15 missiles and a significant number of UAVs and helicopters were destroyed.
Trump has previously vowed to hit back at Iran if it retaliated for the United States drone strike that killed Qasem Soleimani, the head of the IRGC Quds Force. "No Americans were harmed in last night's attack by the Iranian regime".
US Press Secretary Stephanie Grisham released a tweet indicating they are aware of the reports and the President has been briefed.
Iran's Fars news agency called the firing of the missiles at Ain al-Asad airbase a "hard revenge" in a Twitter post.
"The worry is that we could see more from Iran, provoking USA retaliation - a scenario that cannot be ruled out given the warnings from President (Donald) Trump", it said.
Germany, Denmark, Norway and Poland said none of their troops in Iraq were hurt.
More than 5,000 U.S. troops remain in Iraq along with other foreign forces in coalition that has trained and backed Iraqi forces against the threat of Islamic State militants.
General Qassem Soleimani, 62, the head of Iran's elite al-Quds force and architect of its regional security apparatus, was killed when a drone fired missiles into a convoy that was leaving the Baghdad International Airport.
We are aware of the reports of attacks on USA facilities in Iraq.
Moments before news of the rocket attack broke, State Department Spokesperson Morgan Ortagus discussed the current situation in Iran with WSPD host Scott Sands and said that President Trump has shown restraint in dealing with Iranian aggression.
"The president and his crew had better figure out a way to... tone down everything because we could be in the middle of a full-fledged war, and I don't think that is something anybody wants", he said.