The patient entered Iraq before the government shut border crossings and banned the entry of non-Iraqis coming from Iran.
"The results of lab tests conducted on a theology student, an Iranian citizen, who had entered the country before ... the decision to stop travel have revealed he is ill with the coronavirus," a ministry statement said.
The patient is in quarantine and medical staff are observing international health standards, the statement added.
Fears of a coronavirus pandemic grew on Monday after sharp rises in new cases in Iran, Italy and South Korea. The virus has infected nearly 77,000 people and killed more than 2,500 in China, where it originated late last year.
Iraq had earlier on Monday shut its Safwan border crossing with Kuwait to travelers and trade at Kuwait's request, the local mayor told Reuters.
Kuwait and Bahrain on Monday recorded their first coronavirus cases, all people who had visited Iran, state media said.
Iraq on Saturday extended an entry ban for non-Iraqis coming from Iran, which has so far confirmed 43 cases and eight deaths.Neighboring Iraq and Iran, both home to Shi'ite Muslim majorities, share close religious, political, and trade ties.