The six-week conflict between Azerbaijan and ethnic Armenian forces over the region and its surrounding areas was brought to a halt by a Russian-brokered ceasefire deal last month.
The fighting locked in territorial gains for Azerbaijan and has stoked anger in Yerevan, prompting street protests against Armenia's Prime Minister Nikol Pashinyan.
Russian peacekeeping forces have deployed in the region.
Late on Monday, Armenian Deputy Prime Minister Tigran Avinyan said a group of 44 Armenian prisoners had returned to Armenia after Russian mediation, the Interfax news agency reported.
A Russian military aircraft also flew a group of 12 Azeri prisoners to Baku as part of the same swap, Rustam Muradov, the commander of Russia's peacekeeping forces, said in a defense ministry video published on Tuesday.Azeri authorities confirmed their arrival.