Several other Azerbaijani soldiers were captured in the fighting, breakaway Armenian forces said, although Baku has made no comment on the claims.
A 37-year-old Armenian woman, Movses Gasparian, was seriously wounded in the shootout, rebels claimed.
At least 18 soldiers have been killed in a surge in violence along the border since the beginning of the year, with snipers frequently exchanging fire across the frontier.
Most have died in tit-for-tat commando raids. The dead Armenian officer was named as 42-year-old Major Sargis Abrahamyan.
The rebels claimed on Thursday that they had captured another Azerbaijani soldier after a separate raid. Baku, however, said the man named as captured was not a member of their forces.
Armenia-backed separatists seized Nagorny Karabakh from Azerbaijan in a war that left 30,000 people dead as the Soviet Union collapsed. Despite years of negotiations since a 1994 ceasefire, the two sides have yet to sign a peace deal.
Azerbaijan has threatened to take back the majority Armenian-inhabited region by force if negotiations do not yield results, while Armenia has vowed to retaliate against any military action.