Several Armenian and Azeri troops were killed during skirmishes earlier this week as U.S. Secretary of State Hillary Clinton arrived in the region.
The countries went to war over the disputed region of Nagorno-Karabakh in the 1990s. Azerbaijan considers the region to be part of its territory.
A defense minister from the region said one of its forces was killed and two others were wounded in fighting overnight with Azeri forces.
Fighting erupted when "Azerbaijani troops attempted to enter the territory of Nagorno-Karabakh," a statement was quoted by Radio Free Europe/Radio Liberty as saying.
Azeri Foreign Minister Elmar Mammadyarov said the Nagorno-Karabakh conflict was at the top of the agenda during his talks with Clinton in Baku earlier this week.
Talks between Armenian and Azeri officials on the issue are scheduled for June 18 in Paris. Mammadyarov said the meeting was aimed at reaching a "breakthrough in regard to this very longstanding conflict."