Robin van Persie had three good chances to break the deadlock before half-time, but the Gunners lost their way in the second half as the Turkish side grew in confidence.
It was a disappointing result for Arsene Wenger's side as they had beaten Fenerbahce 5-2 in Istanbul only two weeks ago and it came on the back of their 2-1 defeat at Stoke City on Saturday.
Fenerbahce, coached by Luis Aragones who guided Spain to success at Euro 2008 only four months ago, reached the quarter-finals of the Champions League last season but have struggled so far this term and are only ninth in the Turkish League after losing four of their first nine games.
The visitors started very poorly and almost fell behind after only nine minutes when van Persie, sent off at the Britannia Stadium at the weekend, latched on to Cesc Fabregas' delightful chip only to slide his shot wide.
Eight minutes later van Persie missed a sitter when the Dutchman fired a shot into the ground, the ball then bouncing over the bar with the goal gaping.
The north London side continued to dominate possession and went closest to grabbing a winner in the 32nd minute when van Persie's close-range goalbound shot was tipped onto the bar by Demirel Volkan.
The first period had been classic Arsenal with lots of nice crisp passing, but no end product. However, after the break the chances dried up as Fenerbahce started to have more possession and dealt with any sporadic Arsenal attacks with ease.
Early in the second half, Kolo Toure's free-kick was touched over the bar for a corner by Volkan but that was as good as it got for Wenger's side as the match petered out.
The result means that all four English teams in the Champions League have failed to win this week but Arsenal remain top of the group and are still favourites to qualify for the last 16.