The Saints were second best for much of the first-leg game and a deflected goal in each half - the first coming from Alper Potuk with Veysel Sari adding the second - was enough for the hosts to take a decent lead back to McDiarmid Park for the second leg next week.
The atmosphere inside the Ataturk Stadium was electric from kick-off and the hosts nearly repaid that support early on but Patryk Malecki, on loan from Wisla Krakow, fired Veysel's cross just wide of Alan Mannus' right-hand post.
Steve Lomas' side seemed happy to allow Eskisehirspor to retain possession in their own half before closing them down as they attempted to advance.
The home side were well on top and could have taken the lead inside 15 minutes when Veysel headed Dede's free-kick over when well placed.
Former Celtic loanee Diomansy Kamara shot high and wide from outside the box before a goalmouth scramble saw Veysel's attempted shot hit the arm of Saints defender Frazer Wright, but the referee was seemingly unsighted.
Malecki then failed to control the ball inside the area despite being in space as St Johnstone boss Lomas started to prowl the touchline on a more regular basis.
Kamara headed Malecki's deep cross wide of the goal after he had pulled off the shoulder of experienced defender Callum Davidson as Eskisehirspor started to take complete control on proceedings.
Sean Higgins, operating as a loan striker, was becoming more and more isolated as he attempted to feed off long defensive clearances without success.
Although they had not created a single opportunity it looked like the visitors would make it to half-time with the scores level.
But four minutes before the interval Turkey international midfielder Alper collected Kamara's pass and his deflected long-range effort nestled into the corner of Mannus' goal to give Eskisehirspor a deserved lead.
St Johnstone started the second half brighter and forced two corners.
Liam Craig also had two free-kicks, the first almost picking out David McCracken and the second fired straight into the hands of Ruud Boffin in the Eskisehirspor goal.
Alper almost grabbed his second goal in what would have been controversial circumstances.
With Wright down injured at the other end of the pitch, Eskisehirspor launched a counter-attack and only a great save from Mannus kept Alper's effort out.
Full-back Dede played the resulting corner straight out for a goal-kick as a sporting gesture moments after Lomas had been remonstrating with the referee.
The Turkish side did not have to wait long to double their lead however as Veysel was played in on the right-hand side of the penalty area before firing the ball across goal with a heavy deflection off Wright sending the ball past Mannus.
It should have been 3-0 five minutes later when Wright's defensive header fell straight at the feet of Dede but the Brazilian did not show enough composure and Mannus made another decent save.
Lomas introduced Nigel Hasselbaink, Stevie May and Kevin Moon during the second half but they could not provide the spark the Perth side needed to grab an away goal.