Liverpool defender Dejan Lovren skied the decisive penalty as Besiktas triumphed 5-4 on penalties, ending the 1-1 aggregate stalemate.
The blockbuster clash in front of a record 63,324 fans was decided on penalties after Tolgay Arslan levelled the tie with 18 minutes remaining in regulation time.
Arslan then went on to convert Besiktas' fifth spot-kick before Lovren missed his attempt as the Turkish giants avenged their 8-0 Champions League drubbing at Anfield in 2007-08.
"We had a history with Liverpool and I think we have settled the score now," said Cenk.
"It was important for the team, for the club, for the fans.
"[For Lovren's penalty] I waited until the very last moment before he shot. I did not move and I think it played a part in making him miss."
Besiktas team-mate Veli Kavlak added: "We are so happy, I am speechless. It is a very beautiful night for the club. This happiness we feel is indescribable.
"Before the shoot-out, coach [Slaven Bilic] came to us and said he was proud of us – that we had dominated 120 minutes of play and that we had nothing to lose. Then he asked: 'Who wants to take one?' and everybody raised their hand."
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