The draw took place at the Turkish Football Federation headquarters today.
The semi-final games will be played between 28-30 April with the second legs between 19-21 May.
Galatasaray booked a place in the semi-finals after knocking Manisaspor out of the competitions 5-1 on aggregate.
The Lions beat Manisaspor 4-0 at the Türk Telekom Arena but drew the return leg 1-1.
Sivasspor knocked Kayserispor out of the cup in their quarter-final tie.
The Yiğidolar drew the first leg 1-1 and also drew the second game 1-1 before winning the game on penalties.
Galatasaray won the Turkish Cup last season and are currently first in the Süper Lig.
The Istanbul giants are two points clear of second-placed Fenerbahçe and three points off third placed Galatasaray.
Galatasaray will take on Trabzonspor at the Avni Aker stadium on Sunday in the Süper Lig.
The full semi-final draw is as follows:
Galatasaray – Sivasspor
Bursaspor – Fenerbahçe