They began fleeing this week after their forces were routed in the operation by Azerbaijan's military. At least 200 ethnic Armenians and dozens of Azeri soldiers were killed.
On Tuesday, hundreds of cars and buses filled with people and their possessions lined up along the mountain road out of Azerbaijan.
As part of a ceasefire deal, separatists have agreed to surrender their weapons.
The enclave is a part of Azerbaijan that had been beyond Baku's control since the 1991 dissolution of the Soviet Union.
Some Armenians fled, packed into the back of open-topped lorries, others on tractors. Grandmother-of-four Narine Shakaryan arrived in her son-in-law's car, with six people packed inside, Reuters reported.
The 77km drive took 24 hours, she said, and they had not eaten any food.
"The whole way the children were crying, they were hungry," Ms Shakaryan said at the border, carrying her three-year-old granddaughter, who she said had become ill during the journey. "We left so we would stay alive, not to live."
Fuel stations were full on the way out of the Karabakh capital, known as Stepanakert by Armenia and Khankendi by Azerbaijan.
Karabakh's ombudsman said the death toll in the explosion and fire at a fuel depot on Monday had risen to 68, with a further 105 people missing and nearly 300 injured. A total of 68 were taken to medical institutions in Armenia.
The authorities have given no explanation for the blast.
USAID head Samantha Power, in the Armenian capital Yerevan, called on Azerbaijan "to maintain the ceasefire and take concrete steps to protect the rights of civilians in Nagorno-Karabakh".
Ms Power had earlier handed Armenian Prime Minister Nikol Pashinyan a letter of support from US President Joe Biden. She said Azerbaijan's use of force was unacceptable and that Washington was looking at an appropriate response.
Ms Power urged Azerbaijan's President Ilham Aliyev to honour his promise to protect ethnic Armenian rights, fully reopen the Lachin corridor that connects the region to Armenia and let in aid deliveries and an international monitoring mission.
Mr Aliyev has pledged to guarantee the safety of Karabakh's Armenians, but said his iron fist had consigned the idea of the region's independence to history.