The Czech giants raced into a 26-16 advantage early in the second quarter but Tarbes wouldn't allow them to press home the advantage.
Instead, the hosts clawed their way back and took a 41-40 lead early in the second half when Charde Houston knocked down a jump shot.
Eva Viteckova, Taj McWilliams and Petra Kulichova each nailed jumpers for Brno in reply, however, as they went on top at 46-41.
Jan Bobrovsky's team kept its nose in front the rest of the way although Tarbes made them sweat until the end.
After the visitors extended their lead to 10 points in the third quarter, Tarbes continued to fight and the red-hot shooting of Frida Eldebrink helped keep them in the game.
She capped a 9-2 run that bridged the third and fourth quarters with one of her seven three-balls to reduce the deficit to 55-52.
Tarbes twice more pulled to within three points, the last at 59-56.
Hana Horakova buried a jump shot in reply to take her points haul to 14 and recent signing Hollie Grima of Australia followed with a jumper to make it 63-56.
Bobrovsky's side would go on to capture an important road win in a very difficult group.
Sweden international Eldebrink had one of the finest performances of her young career, pouring in 23 points, missing just twice from behind the arc.
Horakova scored 19 points to lead Brno.
Jan Bobrovsky, Brno Coach:
"We had a very hard game, had troubles to stop Eldebrink and Yacoubou-Dehoui, but I think we had more experience in the second half of the game so we could win."
Francois Gomez, Tarbes Coach :
"We lost against a very good team !!! Without experience it was difficult to stop them in the second half."