The first two confirmed cases were announced by the French health minister Agnes Buzyn early on Friday evening.
Ms Buzyn said one patient had been hospitalised in Paris and another in the southwestern city of Bordeaux.
The health minister told a news conference these were the first two confirmed cases in Europe and that more cases were likely to occur in France.
The third was confirmed a few hours later with officials announcing that two of the patients belong to the same family.
Earlier, the charity SOS Medecins said it had treated a patient of Chinese origin who was showing symptoms of a fever and who said he had been in contact with people from Wuhan province in China, epicentre of the coronavirus outbreak.
Ms Buzyn said this patient was 48 years old and had returned two days ago from a trip to China which included a stop in Wuhan.
"He's been put in an insulated room so as to avoid any contact with the outside world. He's fine", she said.
"You have to treat an epidemic as you would a fire, that's to say find the source very quickly," she said. "We identified the first positive cases very quickly."
The Bordeaux patient was in contact with about 10 people before he was taken into care, the minister said. French authorities are seeking to contact them.
The minister urged people who suspect they've sickened to call emergency services and to stay at home to avoid spreading the virus. She said those who came into contact with the two sickened patients would be told likewise.
The newly discovered virus has killed 26 people and infected more than 800.
Most of the cases and all of the deaths so far have been in China, where officials have imposed severe restrictions on travel and public gatherings.
At least 10 cities near Wuhan, where the virus is believed to have originated, effectively putting 33 million people into quarantine.
The virus has created alarm, but there are still many unknowns surrounding it, such as just how dangerous it is and how easily it spreads between people. It can lead to pneumonia, which has been deadly in some cases.
The World Health Organization on Thursday declared the virus an "emergency in China", but stopped short of declaring it a global health emergency.
It comes as the UK Department of Health confirmed that officials are trying to trace 2,000 people who have flown in from Wuhan, China in the past few weeks, in order to check on their wellbeing.
Fourteen people were given the all-clear on Thursday but others are undergoing tests formulated by Public Health England (PHE), Professor Chris Whitty said.
He spoke following a meeting of the Government's Cobra emergency committee in Whitehall, chaired by Health Secretary Matt Hancock.
He said: "We all agree that the risk to the UK public remains low, but there may well be cases in the UK at some stage.
"We have tried-and-tested measures in place to respond. The UK is well prepared for these types of incidents, with excellent readiness against infectious diseases.
"We have global experts monitoring the situation around the clock and have a strong track record of managing new forms of infectious disease."
Prof Whitty said a public health hub would be set up in Heathrow from Friday.
There are no plans to introduce blanket temperature screening of travellers due to the fact it can take seven to 10 days for symptoms of the virus to appear.
One of those cleared of the virus in the UK was Michael Hope, 45, who spent two days in quarantine this week after returning to Newcastle from Wuhan.
As he left the Cabinet Office after the Cobra meeting, Mr Hancock said the risk to the UK public "remains low".
The Chinese city of Wuhan is rapidly building a new 1,000-bed hospital to treat victims, while Disneyland Shanghai and parts of the Great Wall of China have been closed to visitors.
Reuters reported that hospitals in Wuhan are struggling to cope due to medical shortages.
Almost 30 million people and 10 cities in China are now facing travel restrictions.
Earlier on Friday, Professor Paul Cosford, emeritus medical director at Public Health England (PHE), said it was still "early days" in the course of the virus, but stressed that most of those affected abroad are making a good recovery.
But he added it is "highly likely" that cases would be seen in the UK.