The ministry of defence said two people had been killed and a third injured in the incident on Monday.
According to Moscow, the helicopter, an Mi-24, was accompanying a convoy from a Russian military base. Russian forces in Armenia are investigating who is responsible, the ministry said in a statement.
Armenian and Azerbaijani forces have been fighting for six weeks over the region of Nagorno-Karabakh, which is within Azerbaijan but has been under the control of ethnic Armenian forces, backed by Armenia, since 1994.
News of the downed Russian helicopter came as a spokesperson for the president of the separatist government in Nagorno-Karabakh confirmed that Azerbaijani forces had taken control of Shushi, a strategically key city, and were nearing the region’s capital.
The statement on Facebook by spokesperson Vagram Pogosian appeared to confirm a claim the previous day by Azerbaijani president Ilham Aliyev that his country’s soldiers had seized Shushi – the most significant military development in the fighting over the territory.
Shushi’s position above the regional capital Stepanakert, which is about 10 kilometres to the north, gives strategic advantage to whomever holds it. The city also lies along the main road connecting Stepanakert with Armenia.
Long lines of vehicles have jammed the road as Nagorno-Karabakh residents flee the fighting into Armenia.