After blowing their 11-point advantage, Stanislav Eremin's Triumph still had a shot at victory but with three seconds remaining, Sergey Toporov was off target with a three-pointer.
Victory looked assured for Eremin's side with 3:18 remaining after Nenad Krstic's jump shot had given them a 65-54 lead.
Alex Scales gave Kyiv hope with a three-pointer, though, and following Alan Anderson's miss for Triumph, Goran Jeretin drilled a three for Kyiv to close the gap to 65-60.
Neither side scored again until 1:11 remaining in the contest and that's when Kyiv's Scoonie Penn sank a pair of free throws to trim the deficit to just three.
Bremer then turned the ball over back to Kyiv, and Artur Drozdov tipped in Manuchar Markoishvili's miss to make it a one-point game.
Yet another turnover allowed Kyiv to score the go-ahead basket.
Jeretin stole the ball from Kerem Tunceri and drove in for the lay-up and a 66-65 advantage.
Toporov had a chance to win at the end but missed, and arguably the greatest comeback in this season's EuroChallenge belonged to Kyiv.
Kyiv improved to 3-1 in Group H while Triumph are now 2-2.
Scales led the winners with 22 points while Bremer finished with 16 for Triumph.
Sasa Obradovic, coach BC Kyiv:
"First of all I want to congratulate my players. Both teams probably are not happy the way they played tonight, but such an end and such a win must have made our fans happy. Yes, we didn't play well, but a win is a win. We will analyze it a bit later though."
Artur Drozdov, player BC Kyiv:
"We had a good start to the game, but completely stopped. We showed the will to win despite the opponents' big lead. We got back in the game and got the win."
Stanislav Yeremin, coach BC Triumph:
"We played well with the exception of the first quarter, but in the end we got unlucky. Goran Jeretin's huge three and a couple of our mistakes resulted in the loss."
Cerem Tuncery, player BC Triumph:
"If someone is going to look for someone to blame, I am ready to take the blame. I can't afford to make such a turnover during the final possession."