The original match at Fenerbahce's Sukru Saracoglu Stadium on April 26 was abandoned after 57 minutes when Besiktas coach Senol Gunes was hit on the head by an object thrown from the crowd.
Turkey's Football Federation ruled that the game, which was goalless at the point of abandonment and carrying a 2-2 scoreline from the first leg, should resume behind closed doors, but Besiktas had stated that they would not be taking part.
Fenerbahce will be declared winners of the tie by default while Besiktas will not be able to play in the cup next season and will pay back the 4 million Turkish lira (£700,000) that they earned from the competition this year.
Besiktas had tried to overturn the TFF's decision to resume the match, but their appeal was rejected on Wednesday.
Fenerbahce will play Akhisarspor in the Turkish Cup final on May 9.