At least 180,000 people have been forced from their homes by the fighting, which was sparked by an attempted coup on December 15.
The UK cash will pay for emergency medical treatment by Red Cross personnel, as well as providing clean water for nearly 15,000 and tents for up to 7,500 people in one of the world's poorest countries.
Almost half the funds will go to the Rapid Response Facility, which channels money to pre-approved aid organisations and firms that can provide medical, water and sanitation services.
Some £2.5m has been earmarked to fund flights - some organised by the United Nations - to get aid workers and supplies to areas of the country in need.
Ms Greening said: "Many families have fled their homes and are now in urgent need of further help while they seek refuge from the violence. Britain's emergency funding will provide a lifeline to thousands and will ensure aid agencies have the support they need.
"We are monitoring the situation and stand ready to provide further assistance as required."
Britain gave £60m earlier this year to South Sudan - which gained independence in 2011 but has suffered internal conflict since its establishment as the world's newest country.