Keravnos' Jarrett Hart, who represented Great Britain at the EuroBasket this last year in Poland, hit a go-ahead jump shot with just 19 seconds remaining.
After a timeout by Zagreb, the visitors put the ball in the hands of their star center Ante Tomic but the 22-year-old was off target with a potential game-winning jumper.
Zagreb slipped to 2-3 and now need a lot of help to advance to the Last 16.
For them to progress, BC Kyiv must fall at home to Roanne Basket on Tuesday and at Keravnos next week while Zagreb must also prevail in their last game against Roanne.
Should Zagreb and Kyiv finish 3-3, the Ukrainian team would advance on a tie-breaker.
Bryant Smith scored a game-high 29 points and pulled down seven rebounds for Keravnos, with Hart scoring 16 points.
For Zagreb, Tomic had 17 points and nine rebounds and Croatia national team center Nikola Prkácin also had 13 points and nine boards.
Keravnos improved to 1-4 with the success.
Denis Bajamovic, KK Zagreb coach:
"I am a very unhappy man at the end of the night. For the entire game we controlled the rhythm but we were not lucky at the end and lost. We came to Nicosia without three of our starting line-up players. This was a big problem for us and I want to congratulate my players for their effort. If nothing else we still have a chance to qualify."
Petar Babic, KK Zagreb:
"I want to congratulate Keravnos for their win tonight but I think we were the better team for most of the game. At the end we were unlucky and lost by one point. Good luck to Keravnos."
Memos Ioannou, Keravnos Nicosia coach:
"In the last 15 days we have been working very hard to become a better team and improve our defense. This is very important when you play against such good and experienced teams like KK Zagreb which belong to a very different school of basketball. For us it is very important to improve so that we can concentrate on the domestic League. Good luck to K.K. Zagreb."
Andreas Pilavas, Keravnos Nicosia:
"It was a vey good and tough game against a very good team. Except for the first Quarter when we controlled the game the rest belonged to K.K. Zagreb.We tried to minimize our turnovers and I believe that we succeeded to do this; we finished wih only seven turnovers for the game. This helped us to victory in the end. This will also help us for the remaining of the season in our domestic League."