The Turkish side could have won by a greater margin after striking the woodwork three times, including missing a first-half penalty, but had to settle for Korkmaz's 72nd-minute effort.
The hosts came close to making the breakthrough in the 17th minute when Pierre Webo threaded a low ball from the right byline into the path of Moussa Sow but his shot went wide of the post.
Seconds later Dirk Kuyt's cross found Sow, whose header smacked against the bar with the goalkeeper barely twitching.
Benfica got themselves in a good position but Eduardo Salvio got the ball stuck under his feet and his attempted shot merely bobbled across the face of goal.
Maxi Perreira put in a teasing cross which beat the keeper but Fenerbahce managed to clear and Sow found the ball coming to him when he was prostrate in front of goal.
With the game heading into first-half injury-time, Fenerbahce were handed the chance to go ahead when Ola John was adjudged to have brought down Gokhan Gonul.
Although there was contact with the defender's ankle, it appeared minimal and the penalty was harsh. Cristian Baroni's firm shot beat the keeper but also struck the base of the post and ran away from goal, much to the despair of the midfielder who left the pitch in tears, consoled by his team-mates.
Baroni responded by forcing a smart save from Artur with a shot that was sneaking inside the post shortly after half-time, with Raul Meireles then producing a block from the keeper with a fierce shot moments later.
Kuyt then turned and shot but the ball hit the post and bounced into the goalkeeper's arms.
It was Benfica's turn to hit the woodwork when Nicolas Gaitan's curling shot hit the post and bounced away.
Finally, though, Fenerbahce took the lead with 18 minutes to go, although there was some controversy over the award of a corner from which the goal came.
Korkmaz took advantage of Benfica's frustration over the corner to rise highest from the set-piece, nodding in off the bar to send his side into the return leg with a deserved if slender advantage.