The latest clashes are the first serious violation of a Moscow-mediated ceasefire that ended Tuesday the worst outbreak of fighting over the disputed mountainous region of Nagorny Karabakh since it was captured from Azerbaijan by Armenian separatists in the early 1990s.
"Azerbaijan violated a ceasefire overnight... Using mortars" to shell ethnic Armenian rebel positions in Karabakh, the separatist defence ministry said in a statement.
"One (Karabakh) soldier was killed in the northern sector (of the frontline) near Talysh, another one was fatally wounded at the northeastern sector," it said.
"The Armenian villages of Karmir, Ttudjur, and Baganis came under Azerbaijani fire," ministry spokesman Artsrun Hovhannisyan said in a statement.
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Azerbaijan said it returned fire after Armenian forces shelled its positions in Karabakh.
"Azerbaijani armed forces responded to Armenian artillery strikes," defence ministry spokesman Vagif Dargahly said.
"Civilian targets (in Azerbaijan) were also shelled by the Armenian forces."
At least 77 people from all sides, mainly servicemen, have been reported killed since a simmering feud exploded into the worst outbreak of violence in decades on April 2, prompting both Russia and the West to call on the warring sides for an immediate truce.
Armenia dismissed the claims as "untrue".
Armenia-backed separatists seized control of Nagorny Karabakh, a majority ethnic Armenian region inside Azerbaijan, in a war that claimed some 30,000 lives.
Despite a 1994 ceasefire the two sides have never signed a peace deal and sporadic violence along the line of contact regularly claims the lives of soldiers on both sides.
Energy-rich Azerbaijan, whose military spending exceeds Armenia's entire state budget, has repeatedly threatened to take back the breakaway region by force.