Azerbaijan says its forces have "suppressed" the enemy following the incident in the Kalbajar area on the border.
Azerbaijan has said one of its soldiers was killed by Armenian forces, in the latest skirmish in a volatile border region following a 2020 war between the ex-Soviet rivals.
Azerbaijan's Defence Ministry said on Tuesday the soldier died as a result of a "provocation" by Armenian troops and that "all responsibility for tension lies with the military-political leadership of Armenia".
Armenia said later on Tuesday that three of its soldiers had been injured after Azerbaijan deployed drones and artillery against its troops in a region on the country's eastern border.
The Kalbajar district was one of several areas Armenia ceded to Azerbaijan as part of a ceasefire agreement that was brokered by Moscow.
The death follows claims from Armenian-supported separatists in Karabakh earlier this week that Azerbaijani forces opened fire in a village in the region, damaging a car parked near a school.
Defence officials in Karabakh said Russian peacekeepers deployed to the disputed mountainous region had been alerted to the firing, which Azerbaijan's Defence Ministry denied had taken place.
Ethnic Armenian separatists in Karabakh broke away from Azerbaijan as the Soviet Union collapsed in 1991, and the ensuing war claimed around 30,000 lives. The conflict remained largely frozen until the 2020 Karabakh war.
The two Caucasus countries clashed in late 2020 for control of the Armenian-occupied Karabakh region, leaving more than 6,000 dead in a matter of weeks.
During the faceoff, Azerbaijan liberated several cities and 300 settlements and villages that were occupied by Armenia for nearly three decades.