The agencies increased their 2017 appeal for South Sudan's refugees, saying they needed at least $1.4 billion to help alleviate "unimaginable" levels of suffering.
As fighting and hunger drive ever-greater numbers of desperate refugees to flee South Sudan, UNHCR and its partners are today calling on donors to step up funding to provide life-saving aid.
South Sudan has now become the world's fastest-growing refugee crisis, with more than 1.8 million refugees - including one million children - having sought safety in Uganda, Sudan, Ethiopia, Kenya, Democratic Republic of the Congo and Central African Republic.
A World Food Programme (WFP) official called their suffering "unimaginable".
Last month, the UN Office for Coordination of Humanitarian Affairs (OCHA) said 172 people had died across 14 counties in South Sudan since the initial outbreak was reported in June a year ago.
"Aid workers often can not reach the most vulnerable hungry people".
The current rate at which people are fleeing South Sudan exceeds the humanitarian community's already pessimistic estimates. "Many are dying from hunger and disease, many more have fled their homeland for safety overseas", Beasley said, adding that many are "close to the abyss".
Beasley said the number of those displaced by fighting stood at 3.8 million, and that 5.5 million people are facing hunger, while the onset of the rainy season was expected to make many roads unusable, making it harder for help to reach them. "I find that unacceptable, and I hope you do too".
"We can not do a lot of the things we need to do in terms of basic relief and addressing some of the worst consequences of the incredible violence that prevails in South Sudan", Grandi said in a subsequent interview.
Communities hosting refugees are among the world's poorest and are under vast pressure, the world body said in its joint release.
"Helping refugees is not just about providing emergency aid", said UNHCR's Grandi.
UNHCR coordinates the overall response with governments, humanitarian agencies as well as with refugees and host communities. Sudan and Ethiopia are next, with each hosting around 375,000 refugees.
Beasley, who was named WFP chief in March, said he is "deeply saddened" by the misery and starvation suffered by the South Sudanese people.
The UN and its food-assistance branch, the World Food Programme (WFP), had originally requested $1.2 billion but revised the appeal after deciding the amount would not be enough.
According to the U.N panel of experts, recently received documents from a confidential source detailed a contract, signed in June 2014 by two National Security Service officers for a company based in the Seychelles to provide weapons to the South Sudanese Internal Security Bureau, headed by Akol Koor. Each year, WFP assists some 80 million people in around 80 countries.