The commander of the special forces group Tigran Simonyan and a few other soldiers were killed, said the message.
The Azerbaijani armed forces suffered no losses, the situation on the front line is completely under control, the ministry said.
The conflict between the two South Caucasus countries began in 1988 when Armenia made territorial claims against Azerbaijan.
As a result of the ensuing war, in 1992 Armenian armed forces occupied 20 percent of Azerbaijan, including the Nagorno-Karabakh region and seven surrounding districts.
The two countries signed a ceasefire agreement in 1994. The co-chairs of the OSCE Minsk Group, Russia, France and the US are currently holding peace negotiations.
Armenia has not yet implemented four UN Security Council resolutions on the liberation of the Nagorno-Karabakh and the surrounding regions.
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