The dead boy's father, mother, 3-year-old brother and their 6-month-old baby, along with the family's maid, were all repatriated, the minister's statement said.
The outbreak declared in eastern Congo in August has killed more than 1,400 people.
Twenty-seven people who came into contact with the deceased have been placed under observation, the World Health Organization said in a statement.
Three cases of Ebola have been confirmed in Uganda, a neighbouring country to the Democratic Republic of the Congo, officials said on Wednesday June 12.
The woman was the grandmother of a five-year-old boy who died on Tuesday evening after crossing into Uganda with his family from the Democratic Republic of Congo.
According to WHO, Uganda has already administered the Ebola vaccine to around 4700 health care workers across 165 medical facilities.
While awaiting transfer to an Ebola treatment unit, six family members slipped away and crossed into Uganda on an unguarded footpath, authorities said.
For months, officials have feared the virus would cross the heavily travelled border where health workers have been screening millions of people for fever and other symptoms.
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They and frontline health workers are due to be vaccinated Friday with a new drug created to protect them against the virus, the health ministry said. This outbreak "is not going to go beyond" the boy's family in Uganda, he added.
South Sudan has also declared a state of alert and vaccinated health workers.
The emergency committee had in October and again in April held off doing this, partly because the virus remained contained in one part of DRC.
The World Health Organization will hold an emergency meeting Friday to determine whether to declare the outbreak "a public health emergency of worldwide concern", a major shift in the mobilisation against the disease.
Meanwhile, new research from Cambridge University released on Thursday indicates that half of all Ebola outbreaks have gone undetected since the virus was discovered in 1976.
Researchers at the University of Cambridge used several independent datasets from the last major outbreak, in West Africa in 2013-2016, to simulate thousands of outbreaks.
"The unfolding epidemic in the DRC demonstrates how hard it is to stop the disease once it has got out of control, even with global intervention", Emma Glennon from Cambridge's department of veterinary medicine said in a statement.
Ebola is transmitted to people from wild animals and spreads among humans though close contact with the blood, body fluids, secretions or organs of an infected person. Eight others who had been in contact with them were also being monitored in an isolation ward.