Having won the first leg 3-1 at home, Besiktas played calm and collected throughout the game, especially after finding the leading goal in the 25th minute with striker Mustafa Pektemek.
He headed in a corner kick cross by Caner Erkin to give his team the 1-0 lead.
Besiktas' Necip Uysal was shown the second yellow card in the 71st minute, but it was too late for the Ankara team Genclerbirligi to make a comeback against the Istanbul club.
On Thursday, Konyaspor and Galatasaray will look to become the last semifinalist of the cup.
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