The Turkish side were hammered 4-0 in Hungary by Ferencvaros, having lost the play-off first leg at home by a single goal.
Taking full responsibility for the defeat, Tekke said live on air: "I am resigning from a very honourable position.
"Today's defeat is unacceptable. Misfortunes and bad luck are all acceptable, but I cannot accept this. I cannot sleep with this feeling. May everyone forgive me. We did our best."
Before saying: "The team will be much, much better, that's for sure. I won't elaborate further. Nobody knows. As of now, I am resigning from my position after 18 months. I think I have the right to punish myself in this way."
Trabzonspor reject resignation
However, Trabzonspor’s board have confirmed that the resignation was not accepted and that Tekke will remain in control of side heading into the rest of the campaign.
“Fatih Tekke has decided to resign due to the emotional intensity caused by the defeat following the Ferencvaros match,” a club statement read.
“As the Trabzonspor Board of Directors, our Fatih Tekke’s resignation decision has not been accepted.
“Fatih Tekke continues in his role and will be at the helm of our team in the Amedspor match we will play in the league.”
Mohamed Salah, who joined the club this summer alongside former Liverpool teammate Fabinho, will now play in the UEFA Conference League where the side have a serious shot at winning alongside the likes of Ajax, Brighton and Braga.