An Iraqi security official said the rocket struck close to a village outside the base. The official spoke on condition of anonymity in line with regulations.
No one claimed responsibility for Monday's attack, but U.S. officials have previously blamed Iran-backed Iraqi militia groups for assaults against the U.S. Embassy in Baghdad and other military bases hosting American forces.
Recent attacks have been more sophisticated with the use of drones. Earlier this month, a drone targeted Ain al-Asad base causing minor damage.
The killing of Soleimani on Iraqi soil sparked the anger of mostly Shiite lawmakers who passed a non-binding resolution calling for the ousting of U.S. forces from the country.
The future of the U.S. 's military presence in Iraq has been the focus of strategic talks between Baghdad and Washington, with Shiite militia groups insisting on a timeline for withdrawal.