Azerbaijan said one of its soldiers was killed early on March 7 near the ceasefire line that separates Azerbaijani and Armenian forces.
Armenian forces said one of their men had been killed in a separate clash last week.
Both sides blamed the other for triggering the latest exchanges of fire.
The two countries are in a dispute over Nagorno-Karabakh, a mostly ethnic Armenian enclave which split from Azerbaijan in a war in the early 1990s. A 1994 cease-fire ended the fighting but the status of the enclave remains unresolved.
Tensions have been rising since talks in France between the Armenian and Azerbaijani presidents last month in France failed to deliver progress.
(AP, Turan, Reuters)