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.
Galatasaray coach Zafer Kalaycıoğlu:
"This season the important thing is the play-offs. And we are guaranteed to be in the play-offs. So by next week we will know who will be our match up, probably a tough team. I hope it will be good for us. With regard to today's game, my team did their best; at the beginning of the game, only thirty seconds in, we lost Sophia Young and she couldn’t play the rest of the game. It wasn’t good for us of course, but all the other players did a good job in defence. Maybe we were poor in some offensive shots and we lost the game. I hope we have a good oppurtunity to a draw a good opponent for the play-off."
Galatasaray player Tuğba Palazoğlu:
"It wasn’t a really bad game for us. We played good defence but we let them have easy points. We tried to be more agressive in the second half, maybe we could have kept the lead until the end but they had easy shots, open shots and they made them. We lost our consentration and organization and with that the game. They didn’t play as well as I thought they would. We couldn’t use that advantage. And we couldn’t use the advantage of playing at home. I hope next time we play against them we will win."