The mission, led by a senior UN aid official, is the global body's first access to the region in about 30 years.
Armenia has asked the World Court to order Azerbaijan to withdraw all its troops from civilian establishments in Nagorno-Karabakh and give the United Nations access.
The World Court, formally known as the International Court of Justice, in February ordered Azerbaijan to ensure free movement through an area known as the Lachin corridor leading to and from the region.
The process of moving those wishing to relocate from Nagorno-Karabakh to neighbouring Armenia is coming to an end, Russia's RIA news agency quoted the Armenian government as saying late on Sunday.
Earlier on Sunday, the World Health Organisation said well over 100,000 ethnic Armenians from Nagorno-Karabakh have made the journey in less than a week.
"We've activated our emergency systems and will be sending experts to the country across a range of disciplines including mental health, burns management, essential health services, and emergency coordination following a full assessment of the needs," Dr Hans Henri P Kluge, regional director of the WHO Regional Office for Europe, said in a statement.
"The challenges are truly enormous, and we're there to do all we can."
The departure of hungry and exhausted Armenian families this week was blighted by an explosion at a fuel depot that killed at least 170 people.
