Mrs Clinton is the highest-ranking US official to visit South Sudan since it gained independence last July.
A UN deadline for the nations to resolve disputes over their border and oil transit fees passed on Thursday.
South Sudan is the second stop of Mrs Clinton's seven-country African tour.
The row over oil has led to huge economic problems in both countries - South Sudan has suspended all oil production, accusing Khartoum of stealing its exports, while austerity measures have sparked weeks of protests in Khartoum.
"We need to get those [oil] resources flowing again," Mrs Clinton told reporters after talks lasting more than an hour with South Sudanese President Salva Kiir.
"A percentage of something is better than a percentage of nothing," she said, according to the Reuters news agency.
At independence, the South took three-quarters of Sudan's oil with it but all the pipelines still flow north.
"While South Sudan and Sudan have become separate states, their fortunes and their futures remain inextricably linked," Mrs Clinton said.
"Both countries will need to compromise to close the remaining gaps between them."
Ms Clinton's visit to South Sudan came nearly a month after the new state celebrated its first anniversary of independence, which was brought about by a 2005 peace deal between Sudan and the then southern rebels.
The two countries came close to all-out war in April, when South Sudanese troops briefly occupied the disputed oil-rich border area of Heglig.
Negotiations between the two countries in the Ethiopian capital, Addis Ababa, aimed at resolving all outstanding issues, are currently stalled.
Arriving from Senegal via the Ugandan capital Kampala, Mrs Clinton spent only a few hours in Juba before returning to Uganda.
Remarks by Mrs Clinton in Senegal which seemed to criticise China's involvement in Africa were met with a stinging rebuke by the Chinese state media on Friday.
In a speech in Dakar on Wednesday, Mrs Clinton said the United States was committed to "a model of sustainable partnership that adds value, rather than extracts it".
The official Chinese news agency Xinhua said in a commentary: "Whether Clinton was ignorant of the facts on the ground or chose to disregard them, her implication that China has been extracting Africa's wealth for itself is utterly wide of the truth."
Chinese investment in Africa has surged in recent years.
Beijing says it does not interfere in other countries' domestic politics, leading to accusations that it turns a blind eye to human rights abuses and democratic shortcomings.
After returning to Kampala later on Friday, Mrs Clinton met Uganda's President Yoweri Museveni.
"We deeply respect the role that President Museveni has played in his country's history," Mrs Clinton told reporters before leaving South Sudan, the Reuters news agency reports.
However, she also made reference to what critics say is Mr Museveni's increasingly authoritarian behaviour.
"Our position is that there has to be a constitution that sets forth the rules that everyone has to follow... So that it's not about... strong men, it's about strong institutions," Ms Clinton said.
Ms Clinton had also been expected to press Ugandan officials to step up the hunt for the leader of the rebel Lord's Resistance Army, Joseph Kony, and to improve Uganda's human rights record, especially in relation to its often persecuted gay and lesbian communities.
Her 11-day African tour takes her on to Kenya, Malawi, South Africa and Ghana, where she will attend the funeral of the President John Atta Mills on 10 August.
US Secretary of State Hillary Clinton has called on the two Sudans to settle the disputes that have brought them close to war, as she briefly visited the South Sudanese capital, Juba.