The game, which took place at the Basket Hall arena in the capital of Tatarstan, ended with a score of 70:64 in favor of the team of Velimir Perasovich.
Lorenzo Brown was the top scorer for UNICS with 22 points.
Cory Higgins scored 17 points for Barcelona.
UNICS in the match of the 20th round will play with Real Madrid on the road on January 25.
Earlier it was reported that Lokomotiv-Kuban beat Andorra in the Eurocup basketball match.