Looking somber, Armenian Foreign Minister Ararat Mirzoyan and Azerbaijani Foreign Minister Jeyhun Bayramov sat on opposite sides of Blinken at a hotel in New York City ahead of the United Nations General Assembly.
Blinken said he was "encouraged" there had been no violence for several days.
"Strong, sustainable diplomatic engagement is the best path for everyone," Blinken said.
"There is a path to a durable peace that resolves the differences," he said.
The national security council in Armenia revised its death toll from the fighting last week from 136 to 207, taking total fatalities on both sides to 286.
The flare-up last week was the worst fighting since the 2020 war and has jeopardized the nascent peace process between the arch foes.
The new Armenian toll included two civilians, the security council said.
"Two civilians are missing, 293 troops and three civilians were wounded and 20 troops were captured," it added.
Baku has reported 79 deaths among its military.
US House Speaker Nancy Pelosi, during a visit to Yerevan Sunday, blamed Baku for "illegal" attacks on Armenia, condemning an "assault on the sovereignty" of the country.
Washington's ties are deepening with Yerevan whose traditional ally Moscow is distracted with its invasion of Ukraine.
Russia has close ties with both Baku and Yerevan. It is obligated to intervene if Armenia is invaded under a security pact, but did not rush to help despite an appeal from Yerevan.
Armenia and Azerbaijan have fought two wars – in the 1990s and in 2020 – over the contested region of Nagorno-Karabakh, an Armenian-populated enclave of Azerbaijan.
A six-week war in 2020 claimed the lives of more than 6,500 troops from both sides and ended with a Russian-brokered ceasefire.
Under the deal, Armenia ceded swathes of territory it had controlled for decades, and Moscow deployed about 2,000 Russian peacekeepers to oversee the fragile truce.