So far more than 28,000 of the 120,000 Armenians of Karabakh, a region internationally recognised as part of Azerbaijan, have crossed the border into Armenia, a country of about 2.8 million.
Azerbaijan's military victory over the enclave - previously beyond Baku's control - a week ago has triggered one of the biggest movements of people in the South Caucasus since the fall of the Soviet Union.
The hairpin mountain road snaking out of Karabakh towards Armenia was choked with people.
Many slept in cars or searched for firewood to warm themselves by the side of the road.
"I left everything behind," Vera Petrosyan, a 70-year-old retired teacher, told Reuters on Tuesday at the large Soviet-era hotel on the Armenian side of the border with Azerbaijan that is now her home.
"I don't know what is in store for me.
"I have nothing.
"I don't want anything.
"I would not want anybody to see what I have seen," she said, reflecting on the shootings, the hunger, turmoil and suffering she witnessed before escaping to Armenia.
Azerbaijan's 24-hour offensive in Karabakh came amid a blockade imposed on the enclave last December.
It is not clear exactly what happened before the Karabakh leadership agreed to a ceasefire.
Azerbaijan says civilians were not harmed.
Armenia and Azerbaijan have fought two wars over the enclave in 30 years, with Azerbaijan regaining swathes of territory in and around Nagorno-Karabakh in a six-week conflict in 2020.
Azerbaijan President Ilham Aliyev said the rights of Armenians would be respected but said his "iron fist" had consigned the idea of an independent ethnic Armenian Karabakh to history and the region would be turned into a "paradise".
The Armenians of Karabakh told Reuters they did not want to live as part of Azerbaijan and feared ethnic cleansing at the hands of Azerbaijan, which has repeatedly dismissed such claims as nonsense.
Some took down statues of their heroes
As thousands rushed to leave, there was a major explosion at a fuel depot in the Askeran district of Nagorno-Karabakh on Monday, according to local authorities.
It was unclear what caused it.
There were conflicting details about the toll of the blast but the ethnic Armenian authorities said at least 68 had been killed, 105 were missing and almost 300 were injured.
The badly injured were evacuated by helicopter to Armenia as the roads out were so jammed with traffic that a journey of just 77km to the border was taking at least 30 hours.
The Karabakh crisis has shifted alliances in the South Caucasus region, a patchwork of ethnicities wedged between the Caspian Sea and the Black Sea where Russia, the United States, Turkey and Iran vie for influence.
Armenia, which was aligned with Russia, has rowed in public with Moscow which in turn has warned Armenian Prime Minister Nikol Pashinyan, who is facing calls to resign, to stop flirting with the West.
The US, home to the second largest Armenian diaspora in the world after Russia, sent senior officials to Armenia to show their support.
US Secretary of State Antony Blinken asked Azerbaijan's Aliyev on Tuesday "to emphasise the need for Azerbaijan to refrain from further hostilities in Nagorno-Karabakh and provide unhindered humanitarian access", State spokesman Matthew Miller said.
"The Secretary urged President Aliyev to commit to broad amnesty and allow an international observer mission into Nagorno-Karabakh," Miller said.
Aliyev told Blinken "that military facilities only had been targeted during the anti-terror measures, which lasted less than 24 hours, and civilians had not been harmed", according to a statement by Azerbaijan's presidential office.
