Anke De Mondt and Le'Coe Willingham had 19 points apiece to lead Avenida, who won for the sixth time in eight games while Besiktas (2-6) got a game-high 21 from Laura Harper.
Avenida went in front for good when Willingham made both of her three-pointers for a 50-44 lead with four minutes to go in the third quarter.
The advantage grew to 60-50 by the end of the period and Avenida stayed comfortably ahead the rest of the way.
Belgium international De Mondt made all three of her shots from long range.
Willingham also had 11 rebounds while Latvia international Gunta Basko gave Avenida a lift with 10 points and seven rebounds.
Jose Ignacio Hernandez (Avenida coach):
"The key of the game was our defense in the third quarter. We must defend hard for 40 minutes. Now we have two difficult games but we will try to finish in the first place in the group."
Anke De Mondt (Avenida player):
"I´m happy because it was very important to win this game. We have to keep working hard and try to win our next two games."
Aziz Akkaya (Besiktas Cola Turka coach):
"We played hard but in the third quartes we had a lot of turnovers. Congratulations to Avenida. It´s fantastic to play in this gym."