Visitors' Besiktas took the lead in the minute 13 as Turkish midfielder Oguzhan Ozyakup scored the opener at MEDAS Konya Buyuksehir Stadium.
Konyaspor pulled the level four minutes later. Turkish defender Abdulkerim Bardakci put the ball into Besiktas net with a header. The regular time and extra time ended 1-1. So the penalties had to decide the winner.
Besiktas won 3-2 on penalties as Rachid Ghezzal, Valentin Rosier and Necip Uysal converted their penalties successfully.
Only Bernard Mensah missed the penalty for Besiktas.
In Konyaspor, two players scored from penalty spot, Guilherme and Farouk Miya.
Home team's Artem Kravets, Nejc Skubic and Abdulkerim Bardakci missed their shots.
Besiktas will face Medipol Basaksehir in the Turkish Cup's semifinal stage.
The semis will be played as a single leg fixture in March.
Turkish Cup semifinals:
Fraport TAV Antalyaspor - Aytemiz Alanyaspor
Besiktas - Medipol Basaksehir
