The downing of the helicopter belonging to the army of the breakaway ethnic Armenian region is the most serious incident on the Karabakh border since the 1994 ceasefire that ended a bloody war that cost 30,000 lives.
Armenian media reported that the helicopter's three crew members were all killed.
"A MI-24 combat helicopter attempted to attack positions of the Azerbaijani army near Agdam district," Azerbaijan's defence ministry said in a statement.
Yerevan vowed that Baku will face "grave consequences", fuelling fears that the incident might seriously undermine a shaky peace.
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"This is an unprecedented escalation and the consequences for Azerbaijan will be grave," Armenia's Foreign Ministry spokesman, Artsrun Hovannisyan, told AFP.
"Azerbaijan's claim that the Armenian helicopter attacked its positions is not true. Examination of the wreckage will prove that the helicopter carried no weapons," he added.
The separatist defence ministry in Karabakh confirmed that its helicopter was downed by Azerbaijani forces "while conducting a training flight as part of military drills", adding that a firefight began after the incident and was continuing.
Since Thursday, Karabakh forces have been conducting joint drills with Armenia coordinated by the Armenian army chief-of-staff.
Two decades after a ceasefire agreement ended their bitter war over Karabakh, Azerbaijani and Armenian forces regularly exchange fire across their frontier and along the Karabakh frontline.
Last August saw an unprecedented spiral of violence with more than 20 troops killed from both sides in the deadliest clashes since an overall ceasefire was agreed.
Armenian separatists backed by Yerevan seized the mountainous region, which is mainly inhabited by ethnic Armenians, from Azerbaijan in a war in the 1990s.
The peace talks are being brokered by the so-called Minsk group of mediators appointed by the OSCE in 1992 and co-chaired by France, Russia and the United States.