Armenian authorities said on Wednesday they have received 42,500 refugees from Nagorno-Karabakh, following Azerbaijan's lightning offensive last week.
That's more than a third of the separatist Caucasus region's total population.
Baku has promised to guarantee the safety of residents, but Armenia's prime minister warned of "ethnic cleansing".
The mountainous Nagorno-Karabakh is internationally recognised as part of Azerbaijan but was largely controlled by Armenian separatists.
A ceasefire agreement was signed last Wednesday that handed the territory of around 120,000 people to Baku, following its 24-hour military offensive.
On Sunday, Azerbaijan reopened the only road linking Nagorno-Karabakh to Armenia and pledged to allow rebel fighters who surrender their weapons to flee.
The influx of refugees has caused huge traffic jams, with an endless stream of overcrowded vehicles carrying families and their belongings passing through the Lachine corridor.
Many refugees say it took 24 hours to travel the 80 kilometres between Nagorno-Karabakh's de facto capital, known as Stepanakert by Armenians and Khankendi by Azeris, and the Armenian border.
This is often a journey without food and sometimes water, as the separatist region lacks crucial essentials due to Baku's months-long blockade.
“They expelled us,” says one ethnic Armenian who arrived at the frontier on foot.
“I left my house to stay alive,” adds another woman in a green jacket, who insists on speaking: “Let the world know that we are homeless dogs now!”
The Azeris have said they want to treat ethnic Armenians as "equal citizens".
On Monday, Azerbaijani President Ilham Aliev renewed his promise that the rights of ethnic Armenians in the enclave would be "guaranteed".
That same day a massive blast at a fuel depot killed at least 68 people attempting to leave. Nearly 300 more were injured and 105 are missing.
Armenia and Azerbaijan, two former Soviet republics, fought two wars over Nagorno-Karabakh in the 1990s and 2020. The fighting claimed around 36,500 lives.
The death toll from last week's rapid invasion is 200 dead, according to the Armenian side. Dozens of Azerbaijani soldiers were killed as their army poured across the border.
The European Union brought together senior French, German, Azerbaijani and Armenian officials in Brussels on Tuesday.
Talks involved "intense exchanges" over the relevance of a possible future meeting at the start of October.
Top US diplomat Antony Blinken on Tuesday urged Baku to protect civilians in the region and allow in humanitarian aid.
Blinken "spoke again with President Aliyev today and emphasised the urgency of ending hostilities, ensuring unconditional protection and freedom of movement for civilians," State Department spokesman Matthew Miller said.
France called for "international diplomatic action" amid "Russia's abandonment of Armenia."
Paris estimated that the “massive” exodus of Armenians is taking place “under the complicit eye of Russia”, which deployed a peacekeeping force in this region in 2020.