Both Armenia and Azerbaijan accused each other of shelling in breach of a Russian-brokered ceasefire.
The meeting marks the first breakthrough in international efforts to try and broker a ceasefire between the two sides.
Armenia's Nikol Pashinyan and Ilham Aliyev of Azerbaijan discussed the flare-up in violence in Nagorno-Karabakh during exclusive interviews with Euronews.
The three countries issued a joint statement on Monday in which they "France, Russia and the US said they "condemned in the strongest possible terms the dangerous escalation of violence in and outside the Nagorno-Karabakh conflict zone".
Residents in towns along the Azeri-Armenian border are living with the real threat of bombardments as homes and businesses are destroyed by shelling.
Armenian officials on Sunday accused Azerbaijan of carrying out strikes on Stepanakert.
French, Russian and US presidents issue a joint statement urging the leaders of Azerbaijan and Armenia to return to the negotiating table
The official death toll stands at 95, including civilians, as the fighting between Armenia and Azerbaijan continues.
New skirmishes - the deadliest since 2016 — have provoked international concern.
A Europa League match at Luxembourg's Josy Barthel stadium was briefly suspended after a drone carrying the flag of a region disputed by Azerbaijan and Armenia flew onto the pitch.
Ceremonies have taken place in Armenia to mark the anniversary of mass killings and forced deportations of up to 1.5 million Armenians.
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