Yerevan and Baku had fought two wars -- in 2020 and in the 1990s -- over thedisputed Nagorno-Karabakh region before Azerbaijan recaptured in a lightningoffensive last September.
The deaths on Tuesday came several days after the reelection of AzerbaijaniPresident Ilham Aliyev, who has held power for two decades in the oil-richCaspian country.
"Four were killed and one injured as a result of fire on Armenian positionsfrom Azerbaijani troops," the Armenian defence ministry said in a statement.
The fighting occurred near the village of Nerkin Hand in the southern regionof Siunik.
Azerbaijan's border guards said the action was a "riposte" to a "provocation"on Monday by Armenian troops that Baku said had injured one Azerbaijanisoldier.
The Armenian foreign ministry accused Baku of "constantly trying to thwartefforts of parties interested in the stability and security of the SouthCausus" and called for a "return to (peace) talks."
- Calls for restraint -
EU foreign policy chief Josep Borrell slammed Baku's actions.
"The Armenian shooting of the Azerbaijani soldiers yesterday was deplorable.But Azerbaijan's response today seems disproportionate," Borrell said in astatement to the press.
Russia -- which used to be the main mediator of the conflict but has inrecent years been distracted by its Ukraine invasion -- urged calm and calledthe violence "worrying".
"We call on both sides to show restraint to avoid any acts that the oppositeside could see as provocative," Kremlin spokesman Dmitry Peskov said.
Armenia has distanced itself from its historic ally Russia since Azerbaijan'soffensive, with Yerevan angry that Moscow's peacekeeping force in Karabakhdid not act during Azerbaijan's takeover.
The United States, which had brought together the two sides for talks beforeAzerbaijan's military triumph in September, said the latest violenceundermined diplomatic efforts.
"The only way to a sustainable peace is at the negotiating table. Anyceasefire violations should be investigated and properly addressed," StateDepartment spokesman Matthew Miller said.
Baku has denied any territorial claims to Armenia after having regainedKarabakh and ruled out a fresh conflict with Yerevan.
But Aliyev is suspected of trying to wrest control over Armenia's Siunikregion to link Azerbaijan to the exclave of Nakhchivan, which is alsobordered by Iran and Turkey.
Aliyev's election win last week was all but guaranteed after his country'shistoric victory over Armenian separatists last year.
The internationally mediated peace talks have so far failed to produce abreakthrough between Armenia and Azerbaijan.
Armenia's Prime Minister Nikol Pashinyan last month proposed a non-aggressionpact, pending a comprehensive peace treaty between the neighbours, reports BSS.
Pashinyan and Aliyev had previously said a peace agreement could have beensigned by the end of last year.
Almost the entire ethnic-Armenian population of Karabakh -- more than 100,000people -- fled for Armenia following Baku's takeover, sparking a refugeecrisis.