The German team improved to 3-3 with the victory and finished behind Cholet (5-1), while Belgacom slipped to 2-4.
Earl Jerrod Rowland poured in 22 points for Bonn, who trailed 52-47 with 8:09 remaining in the game.
A three-pointer from Moussa Diagne sparked a 9-0 Telekom Baskets run as they claimed the lead.
Belgacom weren't finished, with Christopher Hill making a three-pointer to knot the scores at 56-56 with 2:57 remaining.
Rowland went to the other end and made a three for Bonn and they outscored the home side 11-3 over the last 2:28 of the game.
Will Thomas finished with 11 points and nine rebounds for Belgacom.
First, I want to congratulate Bonn for their win tonight. They played with a lot of energy for 40 minutes and they deserve to win. We came up short at the end of the game, maybe because of our small rotation. Bonn made good offensive decisions during the fourth quarter and their aggressive defence forced us to commit many turnovers. Our shooting percentage was also to low tonght. It's a deception of course, but I think we did all we could in this competition.
Michael Koch, Coach Bonn:
It was a tough game tonight. Liège played with high intensity during the first half, especially defensively. We came back thanks to big 3 point shots from Rowland. We are very happy to being qualified for the Quarter Final Round of EuroChallenge.