Afterwards Clinton travelled to Armenia to meet Serge Sarkisian, the country’s president, who said the dispute over Nagorno-Karabakh was the single most important issue for his country.
His comments echoing almost word for word a statement made hours earlier by Ilham Aliyev, the Azeri president.
“This is a major problem for us and the major threat to regional security,” Aliyev told Clinton.
“We want to find a resolution as soon as possible … our people are suffering.”
Nagorno-Karabakh, which is internationally recognised as a part of Azerbaijan, has been the cause of conflict between Azerbaijan and Armenia for more than a decade.
Tussle over Nagorno-Karabakh town
The conflict reached its peak in the 1990s when majority ethnic Armenians in Nagorno-Karabakh, who sought secession and were backed by Armenian troops, engaged in a war with Azerbaijan over the land, which killed about 30,000 people.
Since the six-year war, periodic armed clashes between both sides have continued.
More than 15 years of mediation efforts have failed to produce a final peace deal between Azerbaijan and Armenia.
Last month four ethnic Armenian troops and an Azeri soldier died in an exchange of fire near Nagorno-Karabakh.
Clinton said that before Armenia and Azerbaijan can move to final negotiations, they need to agree on a basic set of principles for settling the dispute, including a pledge not to use or threaten to use force.
“These are unacceptable violations” of the 1994 ceasefire, she said.
Clinton’s statement came amid strained relations between the US and Azerbaijan, including the absence of a US ambassador, who has been withdrawn from the country for more than a year.
In April Azerbaijan accused the US of siding with Armenia over Nagorno-Karabakh, and threatened to “reconsider” its ties with Washington.
The strains ran so deep that Aliyev received a letter in June from Barack Obama, the US president, saying he was aware of the “serious issues in our relationship” but was confident they could be addressed.
Strategically located between Russia and Iran, Azerbaijan has been a vital supply route for US forces in Afghanistan since 2001.
It also is part of an overland supply chain that is a crucial alternative to the mainland route through Pakistan to Afghanistan.
While seeking to improve relations and make some headway on Nagorno-Karabakh, Clinton also pressed Azerbaijan to show greater respect for civil liberties and raised concerns over two jailed opposition bloggers.