Azerbaijan accuses the Armenian soldiers of trying to cross into its territory in the Kelbajar border region.
Azerbaijan has said that one of its soldiers was wounded after Armenian forces opened fire along the shared border, accusations that Armenia denied.
Friday’s claim marks the latest in a series of incidents between the two ex-Soviet rivals.
In a statement, the Azeri defence ministry said Armenian forces had fired from several directions at its positions in Nakhchivan, an Azeri enclave separated from the remainder of Azerbaijan by Armenian territory.
The soldier was wounded in the shoulder, given first aid and taken to hospital, it said.
Armenia’s defence ministry denied Baku’s assertion, Russia’s TASS news agency reported.
The simmering border dispute comes after last year’s six-week war over the Nagorno-Karabakh region.
The region is internationally recognised as part of Azerbaijan, even by Armenia, but is populated and until recently was controlled by ethnic Armenians.
Friday’s alleged clash came a day after Azerbaijan captured six Armenian servicemen in the Kelbajar district, west of Nagorno-Karabakh.
Armenia said its forces were carrying out engineering work in the area, while Azerbaijan said the soldiers were part of a “reconnaissance and sabotage group”.
Tensions escalated earlier this month, when Armenia accused Azerbaijan’s military of crossing its southern border to “lay siege” to a lake shared by the two countries.
Earlier this week, Armenia said one of its soldiers was killed after shooting broke out with Azerbaijan’s forces, an incident Baku denied responsibility for.
Last year’s conflict, which ended in November, saw Azeri troops drive ethnic Armenian forces out of swathes of territory they had controlled since the 1990s in and around Nagorno-Karabakh.
Russia eventually brokered a ceasefire.
The conflict killed more than 6,000 people on both sides and led to a political crisis in Armenia, where Prime Minister Nikol Pashinyan was widely criticised for what was seen as a humiliating defeat.
Pashinyan, 45, said he had no choice but to concede or see his country’s forces suffer even bigger losses.
He announced snap parliamentary polls under pressure from opposition protesters in the wake of the crisis.
The election is scheduled to be held on June 20.
On Thursday, Pashinyan described the border situation as “tense and explosive”.
Earlier this month, he said Armenia and Azerbaijan were in Russian-mediated talks on the delimitation and demarcation of their shared borders.
He also said the two governments could discuss territorial swaps between the two countries.
Russia’s role as the broker between the two countries has largely come at the expense of Western powers such as France and the United States.
All three are part of a mediating group that had tried but failed for decades to find a lasting solution to the decades-long dispute between Armenia and Azerbaijan over Nagorno-Karabakh.
Armenia fought a war with Azerbaijan over the region in the 1990s which killed at least 30,000 people.