Sudan and South Sudan "will need to compromise to close the remaining gaps between them," Clinton said, after meeting South Sudan's President Salva Kiir.
"It is urgent that both sides, North and South, follow through and reach timely agreements on all outstanding issues, including oil revenue sharing, security, citizenship and border demarcation," she added.
Clinton, the most senior US official to visit since South Sudan's independence last year, highlighted Washington's concern over the bitter dispute between Juba and Khartoum.
South Sudan's government has yet to agree on a raft of issues with Sudan, left unresolved after they split in July 2011, including border demarcation and contested areas in oil-rich regions.
Long running African Union-led talks in the Ethiopian capital have so far failed to produce a deal, with Khartoum rejecting Juba's offers, and demanding that border security must be ensured before any economic accord.
The UN Security Council gave the two countries, which earlier this year came close to a return to all-out war, until August 2 to reach a deal or face sanctions. That deadline elapsed Thursday.
At independence, the land-locked South took with it two thirds of the region's oil, but the pipelines and processing facilities remained in the North.
In January, Juba cut off all oil production, despite it providing some 98% of the South's revenue, crippling both economies, after accusing Khartoum of stealing its crude.
"You have made your point, you have brought Sudan to the negotiating table," Clinton said, standing alongside Foreign Minister Nhial Deng Nhial, stressing the importance of getting the oil to start "pumping again."
"An interim agreement with Sudan over oil production and transit can help address the short term needs of the people of South Sudan, while giving you the resources and the time to explore longer term options."
Clinton warned that "significant challenges" face the world's youngest nation, with "persistent poverty in a land rich with natural resources." Those include "continued violence along the border with Sudan, unresolved ethnic tensions, gaps in infrastructure and the rule of law," she told reporters.
"Continued progress hinges on South Sudan's ability to overcome these challenges," Clinton added.
"While South Sudan and Sudan have become separate states, their fortunes remain inextricably linked," she said.
"The promise of prosperity rests on the prospects for peace. And South Sudan's ability to attract trade and investment depends on greater security on both sides of the border."
Clinton is expected to spend around three hours in the steamy heat of ramshackle Juba -- a rapidly growing city largely made up of simple tin-roof huts strung out alongside the White Nile river -- before flying to Uganda.
"There must always come a point where we look forward and recognize the need to stop fighting over past wrongs so we can build toward a new future," she said.
"It's time...to dig wells instead of graves," she added, quoting a South Sudanese bishop. "Time to reach an agreement that allows both countries to prosper."
Clinton's 11-day, seven-nation Africa tour is focused on the Obama administration's new Africa strategy of promoting development by stimulating economic growth, while advancing peace and security and strengthening democracy.