Lorenzo Brown and John Brown rallied UNICS to victory at Basket-Hall Kazan with 22 and 20 points respectively.
Cory Higgins' 17 points and Brandon Davies' 16 points were not enough for Barcelona to avoid defeat.
The game marked Barcelona's third defeat in their last four EuroLeague games and fifth defeat in the EuroLeague.
Despite this, Barcelona remains atop the EuroLeague standings with 16 wins and five losses while UNICS Kazan are in fourth place with 12 wins and seven losses.
Thursday's Results:
UNICS Kazan-FC Barcelona: 70-64
Zenit St. Petersburg-Monaco: 77-86
Fenerbahce-Zalgiris Kaunas: 73-67
Alba Berlin-Real Madrid: 74-89
Maccabi Playtika Tel Aviv-Olympiacos: 84-69
