Meha gave Konyaspor the lead from the penalty spot on 51 minutes but Medjani equalised just eight minutes later.
It was however, the away side that got the last laugh when Traore scored on 70 minutes.
On-loan Chelsea winger Marko Marin started for the Black Sea Storm and caused Konyaspor problems with his forward runs in fact he completed the highest number of successful dribbles (4) for his team.
Konyaspor moved into third place on 12 points following the win while Trabzonspor dropped into 9th.
TRABZONSPOR 1-2 TORKU KONYASPOR
Stadium: Avni Aker
Referee: Süleyman Abay
Trabzonspor: Alvarado, Cavanda, Mustafa Yumlu, Douglas, Constant, Medjani (min. 81 Özer Hurmacı), Okay Yokuşlu (min. 75 Aytaç Kara), Marin, Erkan Zengin, Yusuf Erdoğan, N’Doye (min. 70 Cardozo)
Manager: Shota Arveladze
Konyaspor: Serkan Kırıntılı, Ömer Ali Şahiner, Vukovic, Ali Turan, Mehmet Uslu, Ali Çamdalı, Holmen, Bajic (min. 70 Uğur İnceman), Traore (min. 81 Ömer Şişmanoğlu), Meha (min. 90 Selim Ay), Rangelov
Manager: Aykut Kocaman
Goals: min. 51 Meha (penalty), min. 70 Traore (Torku Konyaspor), min. 59 Medjani (Trabzonspor)
Red Cards: min. 90+4 Erkan Zengin (Trabzonspor)