The Turkish club hit the woodwork three times including Cristian's penalty which thudded against the post on the stroke of half-time, reducing the Brazilian to tears.
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Luckily for the midfielder, Korkmaz made the breakthrough after 72 minutes with a header that also glanced the frame of the goal before landing narrowly over the line.
It proved enough to put Fenerbahce tantalisingly close to their first European final although Benfica will be formidable opponents in next week's return in Lisbon.
"Normally we are a team that delivers when we find opportunities, but tonight we couldn't do that," Benfica coach Jorge Jesus told reporters.
"But the result is not too bad, we will remain hopeful until the last minute of the second leg. We will be much better in Lisbon."
Roared on by a raucous crowd in the Sukru Saracoglu Stadium Fenerbahce played at a high tempo from the start as Benfica struggled to settle into their stride.
Moussa Sow, who will miss the second leg after receiving a yellow card, struck the crossbar and Fenerbahce maintained the pressure. When Ola John clipped the trailing leg of Gokhan Gonul just before half-time Fenerbahce were awarded a penalty but Cristian failed to convert it.
When former Liverpool striker Dirk Kuyt also hit the post in the second half it looked like being a frustrating night for the home side but Korkmaz raised the roof when he finally broke the deadlock after a poor defensive header.
Benfica's best chance came when Nicolas Gaitan curled a shot against the outside of the post but Fenerbahce held on for the victory despite some anxious moments.
As well as striker Webo, stupidly booked for kicking the ball away, midfielder Mehmet Topal will also miss the second leg after getting booked but coach Aykut Kocaman said his team would fight "until the last drop of their blood" in Portugal.
"It will be a very difficult game for us there, but we definitely have acquired the 51 percent chance of reaching the final," Kocaman said.
Goalkeeper Volkan Demirel was more convincing.
"We will win the cup," he told broadcaster NTV channel after the game.
