The result has dropped Wisla to 0-2.
Jose Ignacio Hernandez's Wisla trailed, 51-40, 2:30 minutes into the fourth quarter but point guard Cristina Ouvina scored to ignite a 9-1 run that was capped by Justina Zurowska.
Györ kept their noses in front but a missed free-throw then proved costly.
Krisztina Raksányi made just one of two attempts at the line with 10 seconds remaining to give her side a 58-55 lead.
De Mondt punished her for the miss at the other end with a three-ball from way out for Wisla to force the extra period.
Zurowska hit a jumper to put the home side in front but Chalysa Shegog leveled the contest and then two Natalie Hurst free-throws put the visitors in front again.
Ouvina knotted the game again at 62-62 with a pair of free-throws but jumpers from Zsófia Simon and Nora Nagy-Bujdosó put the Hungarian outfit back in front to stay.
Dora Horti had a game-high 23 points and 13 rebounds in defeat.
Akos Fuzy, UNI Györ coach:
"I am very happy. We never won here in this gym with my previous team from Pecs. This is my first [victory]. We knew that playing Wisla without their Americans we had a chance to win and so we did it. Thanks to my players for their self-confidence and fighting spirit. Most of teams after that three-pointer of De Mondt, would lose in over-time for sure. [But] we won!"
Krisztina Raksanyi, UNI Györ:
"I am very happy and proud of my team; when Wisla would be complete, with [their] Americans then for sure the Game in Györ will be very good."
Jose Hernandez, Wisla Can-Pack coach:
"We are without Americans and in the last few days without the sick De Mondt and injured Ouvina. Both had to play today - De Mondt being still sick. We also miss Mieloszynska, injured since the begining of the season. We are waiting for Charles and Douglas being the only one club of EuroLeague Women without Americans for the moment. I am proud of the players who fought a lot today and confirming their high sporting level. We are positive for the future."
Anke De Mondt, Wisla Can-Pack:
"I agree with what the coach said. Thanks to all my team-mates for fighting."