The visitors had Raul Meireles sent off early on but were not made to pay, with BATE's finishing wasteful and keeper Volkan Demirel up to the task when he was called upon.
The result surely leaves the last-32 tie balanced in Fener's favour ahead of next week's return leg at the Sukru Sarakoglu Stadium.
Alexander Hleb was a central figure in the early stages, volleying wide in the second minute before, a minute later, he dispossessed Meireles who responded with a frustrated hack from behind that earned him his marching orders.
Demirel earned a yellow card for his protests and was followed into the book by striker Pierre Webo in an ill-disciplined first 10 minutes for the Turkish side.
Hleb had a free-kick saved before Cristian Baroni shot wide at the other end.
BATE were on top but not making the most of their dominance, with Hleb again shooting wide before Demirel saved from Filipp Rudik.
Vitaly Rodionov and Rudik each failed to hit the target with shots before half-time.
Alexander Pavlov replaced Rudik at the break but little else changed, with Alexander Yurevich shooting wide before Baroni had a hopelessly optimistic free-kick blocked for Fener.
BATE's Edgar Oleknovich shot wide and Demirel saved from Pavlov, who sent another effort over the bar.
Vitaly Rodionov fired across goal from wide on the right and then drew a sharp close-range block from the impressive Demirel with the best chance of the game as the home side began to sense the breakthrough would not come.
Alexander Volodko had two unthreatening efforts as time ran down, a weak long-range volley held by Demirel and a shot over the bar.