A strong start to the game was the key for the Turkish giants as they gave the home team a mountain to climb after registering a 17-4 run in the first half to move from 12-11 in front to 29-15.
It looked like the home team weren't going to make a run at all in the game until the last four minutes when they suddenly sparked into life and gave their opponent something of a scare.
Trailing 55-45, three-pointers by Jami Montagnino and Michelle Greco put them to within four points with 46 seconds remaining in the game.
However a big defensive play by Penny Taylor helped seal the win for Fenerbahce as she produced a huge block in the closing moments.
The Australian followed up her classy performance in the first game with another pivotal contribution of 17 points and 5 rebounds while Nicole Powell claimed 15.
Rebekkah Brunson finished with 10 points and 8 rebounds for the home team who will be cursing their slow and costly start to the game.
Haydar Kemal Ates, Fenerbahce coach:
"I am happy for this success. We deserved it and the 2-0 scoreline highlights our superiority. Now we focus on the mighty Spartak Moscow Region."
Valentina Siccardi, Cras Basket Taranto player:
"It 'was a good game, but we lost. We need to apply more pressure; but the opponent was perfect. We leave this season's EuroLeague but with our heads held high."
Devran Tanacan, Fenerbahce Player
"We are happy for this victory, because it let us keep our European dream alive. Today we have proved to be a unified and alive group."