Azerbaijan says one of its generals and five other officers have been killed in a third day of fighting with Armenian forces on the countries' border.
An Azeri private also died, bringing the Azeri death toll so far to 11. Armenia says four of its troops - two of them officers - were killed.
Both countries were part of the Soviet Union until its collapse in the 1990s.
They fought a bloody war over a mountainous territory, in a dispute that remains unresolved.
Nagorno-Karabakh is internationally recognised as part of Azerbaijan but is controlled by ethnic Armenians.
This clash, however, took place north of this disputed territory.
Azerbaijan says heavy fighting is continuing in Tovuz district, bordering on Tavush in north-eastern Armenia.
Dmitry Peskov, spokesman for Russian President Vladimir Putin, said Russia was "deeply concerned" about the outbreak of violence and was willing to mediate.
"We urge both parties to show restraint and to comply with their obligations under the ceasefire."
A US state department release on Monday said the country "condemns in the strongest terms the violence along the Armenia-Azerbaijan international border" and called on both sides to stop the fighting and abide by the ceasefire.
What's happening on the border?
Both sides accuse each other of shelling civilian areas on the border.
Azerbaijan's defence ministry said a 76-year-old man was killed in the village of Agdam, by Armenian shelling.
Earlier the ministry said four Azeri troops had been killed in clashes on Sunday and Monday, involving tanks and artillery.
Meanwhile, Armenia accused Azerbaijan of shelling the town of Berd.
Azerbaijan's military said it had destroyed an Armenian fortification and artillery and had inflicted casualties on "hundreds" of Armenian troops. Armenia denied suffering any casualties on that scale, but reported several wounded besides the two officers killed.
Among the six Azeri officers killed were Maj-Gen Polad Hashimov and Col Ilgar Mirzayev.
Armenia named its dead officers as Maj Garoush Hambardzumyan and Capt Sos Elbakyan. Later the defence ministry said two Armenian sergeants had been killed too.
Gas supplies to three Armenian villages were cut when some local pipelines were damaged by shelling, Gazprom Armenia said.
At an emergency meeting on Monday, Azerbaijani President Ilham Aliyev said: "Armenia's political and military leadership will bear the entire responsibility for the provocation."
Armenia, however, accused its neighbour of "using artillery in an attack aimed at capturing [Armenian] positions".
"This aggression against the security of the civilian population of Armenia will receive a proportionate response, for which the Azerbaijani side bears full responsibility," an Armenia foreign ministry spokeswoman said in a statement.
A collapse in diplomacy
By Rayhan Demytrie, BBC Caucasus correspondent
The escalation in fighting comes just days after Azerbaijan's President Aliyev criticised international mediators conducting peace negotiations with Armenia, describing the process as "meaningless". He also accused Armenia of dragging its feet in order to maintain the status quo.
Azerbaijan is frustrated that after nearly three decades there has still been no progress towards settling the conflict over the breakaway region of Nagorno-Karabakh, and the return of seven adjacent Azeri territories currently under Armenian control.
Known to Armenians as Artsakh, the self-proclaimed republic in Nagorno-Karabakh has been insisting on its right to self-determination.
Armenia's Prime Minister Nikol Pashinyan proposed including the separatist government in peace negotiations with Azerbaijan. But that is unacceptable to Baku, which says the dispute is with Armenia alone.
The sides accuse each other of starting the latest flare-up, and all eyes are now on Russia. It helped negotiate a ceasefire in 2016 after the so-called "April War" - in which some 200 soldiers and civilians were killed, and the two sides came close to all-out war.
What's the history?
The two ex-Soviet republics in the Caucasus fought a bitter war in the 1990s, when Armenia backed the majority ethnic Armenians in Nagorno-Karabakh. A fragile ceasefire was agreed in 1994.
Tensions last flared into conflict in 2016, with the countries clashing over the disputed territory for four days. The BBC reported from both sides of the fighting.
The Organisation for Security and Co-operation in Europe (OSCE) has long been trying to mediate a settlement of the conflict, with diplomats from France, Russia and the US - making up the OSCE Minsk Group - trying to build on the ceasefire.