The Greek side took the lead in the fourth minute through Marko Livaja, but the Turkish side had other ideas when Ekuban levelled matters after 29 minutes thanks to a Filip Novak assist.
Unal Karaman’s men went into the halftime break with a 2-1 lead after Jose Sosa assisted the Ghana international for his second of the day.
However, all hopes by the hosts to launch a comeback faded into thin air as Ekuban completed his treble in the 70th minute, with Abdulkadir Omur getting the assist.
The 25-year-old was given a standing ovation by the fans after coming off for Yusuf Sari in the 90th minute.
Former Nigeria international John Obi Mikel was not used by manager Karaman, while his compatriot, Anthony Nwakaeme lasted for 71 minutes for his replacement by Donis Avdijaj.
Trabzonspor host the Yellow-Blacks in the play-off return leg on 29 August at Athens’ Olympic Stadium.