Poland international Bibrzycka scored 16 points while USA international Cappie Pondexter poured in a team-high 20 for UMMC.
Galatasaray's Jia Perkins led all scorers with 23 points.
The loss sealed a fourth-place finish for the Turkish outfit.
USA superstar Tamika Catchings is expected to join Galatasaray later this month.
With her in the line-up against UMMC, Zafer Kalaycioglu's team might have had enough firepower to get the win.
The key stretch in the game occurred early in the final period.
The home side went in front at 55-52 on Esra Sencebe's three-pointer.
UMMC's recent signing Candace Parker replied with a lay-up before Biba's go-ahead three-ball.
Celine Dumerc then hit the Turkish team with another three-point dagger before Sandrine Gruda split a pair of free-throws for a 61-55 advantage with 6:38 to go.
UMMC stayed on top but weren't able to deliver the knockout punch and with 30 seconds left, they only led 66-64 following a Perkins lay-up and a Yasemen Saylar jumper.
The visitors called a timeout and when play resumed, Gala fouled Pondexter who sank both free-throws with 20 ticks remaining.
Tugba Palazoglu next missed from three-point range for Galatasaray and Pondexter grabbed the rebound to seal victory for UMMC.