The Saints were second best for much of the 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.
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 Diomansy Kamara's pass and his deflected long-range effort nestled into the corner of Alan Mannus' goal to give Eskisehirspor a deserved lead.
The Turkish side did not have to wait long after the break to double their lead 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 Frazer Wright sending the ball past Mannus.
Other British and Irish sides in action also fared poorly on Thursday.
Ireland's Sligo Rovers lost 3-1 at Spartak Trnava of Slovakia, while Northern Ireland's Portadown were thrashed 6-0 by Croatian side Slaven Belupo.
But there was a decent result for Ireland's St Patrick's Athletic, who drew 1-1 at Bosnia's Siroki Brijeg.
Other key matches saw Steve McClaren's FC Twente surprisingly held to a 1-1 draw by Finnish side FC Inter Turku, while another Dutch side Vitesse Arnhem fought out a thrilling 4-4 draw at Bulgaria's Lokomotiv Plovdiv.