Turkish media say the attacker is a research fellow at the city's Osmangazi University and has been detained.
The shooting killed the deputy dean, faculty secretary, lecturer, and a staff member from the education faculty of the university.
Hurriyet newspaper said three other people were wounded in the attack.
Police have arrested the suspect, thought to be a research assistant, and taken him into custody.
"In this unfortunate event, vice dean Mikail Yalcin, faculty secretary Fatih Ozmutlu, teaching faculty Serdar Caglak and researcher Yasir Armagan lost their lives".
Eskisehir, a city of around one million people, has been mostly spared of violence of any kind.
Media reports said the gunman, identified as Volkan D., turned himself in to police after the attack. Students and teachers panicked as images showed them gathering outside the main entrance of the university while ambulances were reaching the area.
Turkey has been under a state of emergency since the coup attempt.
Gonen said the gunman's claim was being investigated by authorities prior to Thursday's shooting.