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.