Konyaspor defeated Trabzonspor 1-0 at home in the first leg on December 27, 2017.
Omer Ali Saliner opened scoring for Konyaspor in the 24th minute.
With 10 minutes remaining, Hugo Rodallega scored for Trabzonspor to make it 1-1.
Trabzonspor needed to score two more goals but Konyaspor held firm to book their place into the quarter-finals.
Earlier on Tuesday, Super Eagles midfielder Mikel Agu was on target for Bursaspor who also crashed out of the Turkish Cup despite defeating Genclerbirligi 2-1.
Bursaspor were eliminated via the away goals rule following their 1-0 first leg loss to Genclerbirligi.
Also in action for Bursaspor was William Troost-Ekong.