On one of the roads leading to Armenia, the destination of those leaving, a column of hundreds of cars was formed, the BBC reports.
Azerbaijan says residents will be safe, but Armenia's prime minister claims "ethnic cleansing" has begun.
Nagorno-Karabakh, recognized as part of Azerbaijan, has been run by ethnic Armenians for three decades, according to the BBC.
The UN Security Council scheduled an emergency meeting on Wednesday to discuss the worsening humanitarian situation in Nagorno-Karabakh, or Artsakh, as a result of Azerbaijan's blockade of the Lachin corridor. For over eight months, the region's 120,000 indigenous Armenians have been deprived access to food, medicine, fuel, electricity, and water.
Türkiye on Sept. 13 firmly sided with its regional ally Azerbaijan in the latest deadly outbreak of violence in the Caucasus, telling Armenia to "cease its provocations" against Baku.
Clashes erupted overnight along the volatile Azerbaijan-Armenia border near the occupied region of Nagorno-Karabakh, leaving troops dead on both sides, defense ministries in Baku and Yerevan said.
President Recep Tayyip Erdoğan has extended his congratulations to Azerbaijani President Ilham Aliyev on his country's control of Lachin and other cities in a phone conversation.
According to a statement by the Presidential Communications Directorate, Erdoğan spoke on the phone with Aliyev over the weekend to address the bilateral relations and recent regional developments.
New tensions erupted over the occupied Nagorno-Karabakh on Aug. 3 as three soldiers were killed and Azerbaijan said it had taken control of several strategic heights in the disputed region.
Arch enemies Armenia and Azerbaijan fought two wars - in 2020 and in the 1990s - over the occupied region of Nagorno-Karabakh.
Turkey and Azerbaijan on Sept. 6 launched a joint military drill in Azerbaijan's Lachin region that was liberated from Armenian occupation last year, according to the Azerbaijani Defense Ministry.
The joint exercise is being held under the Agreement on Military Cooperation between Azerbaijan and Turkey, the ministry said in a statement.