The defence ministry in Baku said "Armenia's reconnaissance and sabotage groups had once again tried to attack Azeri positions at the line of contact" near the majority Armenian region of Nagorny Karabakh.
Azeri troops repelled the overnight attack, forcing Armenian soldiers to retreat, the defence ministry said.
"As a result of the clash, four Azeri troops died," the ministry said in a statement.
Nagorny Karabakh for its part said it had lost a soldier, a 25-year-old ethnic Armenian, and accused Azerbaijan of attempting "sabotage and reconnaissance activities".
The latest clashes came after Azerbaijan said yesterday it had lost eight soldiers in three days of skirmishes with Armenian troops on the border and near the disputed region.
The two ex-Soviet nations have for years been locked in a protracted conflict over Nagorny Karabakh with occasional skirmishes along the front.
But the latest clashes represent a surge in tensions, with one prominent Azeri military expert saying Baku has not suffered such losses in a single bout of hostilities since 1994.
"Retaliation and further violence will only make it more difficult to bring about a peaceful settlement," deputy State Department spokeswoman Marie Harf said in a statement.
Armenian-backed separatists seized Nagorny Karabakh from Azerbaijan in a 1990s war that killed 30,000 people.
Despite years of negotiations since a 1994 ceasefire, the two sides have yet to sign a peace deal.
Energy-rich Azerbaijan has threatened to take back the disputed region by force if negotiations do not yield results, while Russia's ally Armenia has vowed to retaliate against any military action.
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