The hosts made a quick start to the match in Istanbul when new signing Nani, who arrived from Manchester United this summer, tested Andriy Pyatov with a stinging drive from 20 yards.
Shakhtar were the next to threaten, however, doing so 10 minutes later when Alex Teixeira put his effort narrowly past the post having been found unmarked inside the area.
The visitors continued in search of a valuable away goal to take back to Ukraine, and Taison almost provided it with just over 15 minutes remaining in the half, only to be denied by Volkan Demirel.
Fenerbahce were without Robin van Persie, but his former United teammate Nani continued to look the most likely for the hosts, lifting the ball into the path of Fernandao, whose header flew over the crossbar.
That proved to be the last meaningful chance of the half as both sides were forced to go into the break all square and goalless.