Officials in Baku and Yerevan confirmed shelling with large-calibre weapons and said fighting was ongoing near the border in the Tavush region.
Russia has said it is prepared to mediate peace talks between ex-Soviet rivals Azerbaijan and Armenia after fighting escalated along their shared border.
President Vladimir Putin and members of Russia's Security Council expressed "extreme concern" over the flare-up in fighting between the South Caucasus countries that erupted last week, Kremlin spokesman Dmitry Peskov said.
Putin alongside senior Russian government officials "emphasised an urgent need for the parties to adhere to a ceasefire, and voiced a readiness to mediate efforts", Peskov said, quoted by the Interfax news agency.
Turkish defence industry says it can support Azerbaijan
Turkey's defence industry chief said on Friday that his sector was ready to help Azerbaijan, which has seen border clashes with Armenia in which 16 people have been killed.
Turkey has strong historical and cultural ties with Azerbaijan, as well as joint energy projects.
President Recep Tayyip Erdogan said on Tuesday that it would not hesitate to "stand against any attack" on Azerbaijan and that Armenia was "out of its depth" in the conflict.
On Friday, Defence Industry Director Ismail Demir tweeted that "Our defence industry, with all its experience, technology and capabilities, from our armed drones to our ammunition and missiles and our electronic warfare systems, is always at the disposal of Azerbaijan!".
Fifteen soldiers from both sides and one civilian have died since Sunday in the clashes between the neighbouring former Soviet republics, which fought a war in the 1990s over the mountainous Nagorno-Karabakh region.
International concern is high because of the threat to stability in a region that hosts pipelines taking oil and gas from the Caspian Sea to global markets.
Demir, who met Azerbaijan's deputy defence minister and air force commander Ramiz Tahirov in Ankara, said Turkey would help to modernise Azerbaijan's army.
Turkish Defence Minister Hulusi Akar said Armenia had started the border clashes, and that it would be "drowned under the plot that they initiated".
Fresh clashes at Azerbaijan-Armenia border
Border clashes between arch-foes Azerbaijan and Armenia have resumed after a brief de-escalation in fighting.
Azerbaijani defence ministry said one of its soldiers died, while Armenian defence ministry said a civilian was wounded in Chinari village from an Azerbaijani drone strike.
Azerbaijan said early on Thursday clashes were ongoing near the border after "Armenians shelled Azerbaijani villages with large-calibre weapons" and Armenia's defence ministry said Azerbaijani forces were "shelling Armenian villages with mortars and howitzers."
Defence officials a day earlier said fighting had subsided in the Tavush region after several days of deadly clashes.
Azerbaijan said it has lost 12 servicemen and one civilian in three days of fighting, and Armenia said four of its troops were killed on Tuesday.
On Thursday, Armenian officials said 20 servicemen have been injured since Sunday, with one of them in grave condition.
The fighting had prompted calls for an immediate ceasefire from the United States, European Union and regional power broker Russia.
Internationally mediated peace talks between the two Caucasus nations have so far failed to bring about a solution to the territorial dispute.
Azerbaijani minister sacked
Azerbaijan's President Ilham Aliyev sacked his foreign minister after accusing him of "meaningless negotiations" with Armenia.
Aliyev said on Wednesday Foreign Minister Elmar Mamedyarov did not do enough to try to resolve the conflict.
"What was the foreign minister doing? Where he was? We were all at work after the July 12 events ... and I could not find him," Aliyev told a government meeting of the start of the clashes on Sunday.
"... Unfortunately, recently our diplomacy is not compatible with the successful development of our country. In some cases, it is engaged in meaningless work, meaningless negotiations."
Mamedyarov, 60, has been foreign minister since April 2004.
Locked in conflict
The two neighbours in the South Caucasus have been locked in conflict over Nagorno-Karabakh, a region of Azerbaijan that has been under the control of ethnic Armenian forces backed by Armenia, who declared independence during a conflict that broke out as the Soviet Union crumbled in 1991. International efforts to settle the conflict have stalled.
The latest clashes occurred some 300 km from the mountainous enclave.
Though a ceasefire was agreed upon in 1994, Azerbaijan and Armenia continue to accuse each other of shooting attacks around Nagorno-Karabakh and along the separate Azerbaijani-Armenian frontier.
Armenian and Azerbaijani forces have frequently engaged in clashes. In 2016, scores were killed in four days of fighting.
Turkey slams Armenian aggression
Turkey's Foreign Ministry slammed Armenia's attacks on Azerbaijan and the smear campaign against Turkey.
"This hypocritical attitude of Armenia, which has maintained an illegitimate occupation in the territory of Azerbaijan for many years, clearly and obviously reveals who is the main obstacle to the establishment of permanent peace and stability in the South Caucasus," it said in a statement.
The Foreign Ministry said Armenia's foreign policy based on slander will not be beneficial to anyone.
“This approach is a manifestation of a mentality that creates its identity only from a unilateral understanding of history and tries to justify its unlawful aggression,” it said.
The ministry added that the Armenian authorities should act wisely and learn as soon as possible to be part of solutions instead of problems in the South Caucasus.