Since it emerged in December, the virus has killed 2,345 people in China, the epicenter of the epidemic, and 16 elsewhere in the world.
The COVID-19 outbreak in Iran first surfaced on Wednesday, when authorities said it claimed the lives of two elderly people in Qom, a holy city south of the capital.
They were the first confirmed deaths from the disease in the Middle East.
"We have 10 new confirmed cases of COVID-19," Iran's Health Ministry spokesman Kianoush Jahanpour told national television on Saturday.
Jahanpour said that of the 10 newly detected cases, two were in the capital of Tehran and eight were in the city of Qom. He said the two patients in the capital had visited Qom or had links with the city.
"One of the new cases has unfortunately passed away," he added, noting that eight of them had been hospitalized in Qom and two in Tehran, without specifying where the death occurred.
The latest cases take to 28 the total number of confirmed coronavirus infections in Iran.
The outbreak emerged in the lead-up to a parliamentary election on Friday. State media said the disease had failed to dampen "the revolutionary zeal of Qom's people" to turn out to vote.
Iranians have been snapping up surgical face masks in a bid to avoid catching the virus.
The Health Ministry said tests had been carried out on 785 suspected coronavirus cases since the outbreak began.
"Most of the cases are either Qom residents or have a history of coming and going from Qom to other cities," its spokesman said.
All of those who lost their lives are believed to be Iranian citizens.
Iran has yet to confirm the origin of the outbreak, but one official speculated that it was brought in by Chinese workers.
"The coronavirus epidemic has started in the country," IRNA quoted the Health Ministry's Minoo Mohraz as saying.
"Since those infected in Qom had no contact with the Chinese ... the source is probably Chinese workers who work in Qom and have traveled to China," she added.
A Chinese company has been building a solar power plant in Qom.
Qom is a popular religious destination and a center of learning and religious studies for Shia Muslims from inside Iran, as well as Iraq, India, Lebanon, Pakistan, Afghanistan,L and Azerbaijan.
Concerns over the spread of the virus, which originated in central China, prompted authorities in Iran to close all schools and seminaries in Qom.
Iran had recently evacuated 60 Iranian students from Wuhan, the Chinese city at the epicenter of the epidemic. The students were quarantined upon their return to Iran and were discharged after 14 days without any health problems.
Iran has suspended all passenger flights with China for the past two weeks, allowing only cargo flights.
Iran’s civil aviation spokesman Reza Jafarzadeh said on Thursday that the “cargo flights, if necessary, are under supervision, and controls imposed by the Health Ministry are carried out.”
Following the announcement of the deaths, neighboring Iraq on Thursday banned travel to and from the Islamic Republic.
The Iraqi Health Ministry announced that people in Iran were barred from entering the country "until further notice."
Kuwait's national carrier Kuwait Airways also announced it would suspend all flights to Iran.
In Turkey, Health Minister Fahrettin Koca said officials have started to screen travelers arriving from Iran at border gates and are refusing entry to anyone with signs of illness. He also said Iranians who have traveled to Qom in the past 14 days will be refused entry.
Saudi Arabia announced that citizens and residents of the kingdom are not permitted to travel to Iran following the spread of the virus there. Anyone previously in Iran will only be permitted entry to the country after the 14-day incubation period of the virus has passed.
The decision, while not specifically mentioning Iranian nationals, directly impacts thousands of Iranians who travel to Mecca and Medina for Islamic pilgrimages, effectively barring them from entry to Saudi Arabia.
AFP and AP contributed to this story.