"In the 21st century, it is utterly despicable that sexual exploitation and abuse continues to exist in the aid sector", she sad.
Mordaunt spoke Sunday - three days after a Times of London investigation accused Oxfam's then-director in Haiti, along with other workers, of running an illegal makeshift brothel after a 2010 quake devastated the country.
She said her "absolute priority" was to keep the world's poorest and most vulnerable people safe from harm.
A probe was announced after it emerged Oxfam employees romped with vulnerable sex workers following a devastating natural disaster. The charity allowed staff members to resign from the organisation and seek new aid jobs elsewhere.
Ms Mordaunt has written to all United Kingdom charities which receive United Kingdom aid to tell them they must declare all safeguarding concerns or lose Government funding.
Late on Friday, the Department for International Development (DFID) also said it was reviewing its relationship with the UK-based charity, to which it gave almost £32 million (36 million euros/$44 million) previous year.
"This is a historic case but it is in some respects still live".
'We also want to satisfy ourselves that we do now have a culture of openness and transparency and that we fully learn the lessons of events in 2011'.
She has written to all organisations funded by her department, telling them to report any safeguarding issues.
Mark Goldring said the UK-based charity was ashamed of what it had got wrong but had taken action and been proactive about going public on the matter.
Marr said: "That was a lie, wasn't it?"
Asked by Marr if she thought Oxfam had failed in its "moral leadership", Ms Mordaunt replied: "Yes, I do".
"If the moral leadership at the top of the organization isn't there, then we can't have you as a partner", she said.
"There are some actions I'm looking for them taking now".
"If they do not hand over all the information that they have from their investigation. then I can not work with them any more as an aid delivery partner - and any other organisation in those circumstances", she told the BBC.
Hundreds of people, many of them Haitian, demonstrate against racism in Times Square on Martin Luther King (MLK) Day on January 15, 2018 in New York City.
Oxfam was further embroiled in scandal as reports claimed staff also used prostitutes in Chad during a humanitarian mission there in 2006.
Oxfam has said it was "dismayed by what happened" and has since set up a hotline for people to report sexual abuse and misconduct.
Save the Children recorded 31 incidents - 10 referred to the police and civil authorities - and Christian Aid recorded two.
The British Red Cross admitted there had been a "small number of cases of harrasment reported in the UK", believed to be as many as five.