A Russian military helicopter crashed in Armenia Monday after it was shot down by Azerbaijan, killing two crew members, an incident that could prompt the Kremlin to depart from the strict neutrality it has sought to display since the two former Soviet republics resumed a war for control of a disputed territory.

The helicopter, an Mi-24 gunship, was flying near Armenia’s border with Azerbaijan, escorting a convoy from a Russian military base in Armenia, when it came under fire, according to a statement from Russia’s Defense…