The two countries have fought two wars for control of Azerbaijan's Armenian-populated enclave that have claimed tens of thousands of lives.
Internationally mediated peace talks between the ex-Soviet republics have since produced little, if any, result.
But Armenian Prime Minister Nikol Pashinyan on Thursday said he had presented to arch-foe Baku a project for a full peace treaty to end the Caucasus neighbours' dispute.
"We believe that Armenia and Azerbaijan have a genuinely historic opportunity to secure an enduring peace after more than 30 years of conflict," Blinken said ahead of a meeting with Pashinyan and Azerbaijani President Ilham Aliyev on the sidelines of the Munich Security Conference.
"The parties themselves have renewed their focus on a peace process, including through direct conversation as well as with the EU and ourselves," Blinken said.
The United States remains "committed to doing anything we can to support these efforts", he added.
When the Soviet Union collapsed in 1991, ethnic Armenian separatists in Karabakh broke away from Azerbaijan. The ensuing conflict claimed around 30,000 lives.
Another flare-up in violence in 2020 left more than 6,500 dead and ended with a Russian-brokered truce that saw Armenia cede territories it had controlled for decades.
Pashinyan's announcement about the peace treaty came after Yerevan accused Baku of conducting a "policy of ethnic cleansing" and forcing ethnic Armenians to leave the breakaway region.
Since mid-December, a group of self-styled Azerbaijani environmental activists has barred the only road linking Karabakh to Armenia to protest what they say is illegal mining.
In a statement after the meeting in Munich, Pashinyan's office confirmed that the draft peace treaty had been discussed.
"Pashinyan reaffirmed the determination of the Armenian side to achieve a treaty that will truly guarantee long-term peace and stability in the region," his office said.
However, he had also denounced "Azerbaijan's illegal blockade of the Lachin Corridor and the resulting humanitarian, environmental and energy crisis in Nagorno-Karabakh", the statement said.
Aliyev told journalists after the meeting that it had taken place "in a constructive manner", and he was "studying" the Armenian proposals, according to the RIA Novosti news agency.
"At first sight, there is progress regarding Armenia's position, but it is not enough," he said.