St Johnstone briefly threatened an upset after Gregory Tade's debut goal but former Celtic striker Diomansy Kamara created the strike that killed off their chances.
Former Inverness forward Tade headed home in the latter stages of a first half of few chances to give the Perth side hope of overturning Eskisehirspor's 2-0 first-leg lead.
Nigel Hasselbaink came close to levelling matters but Kamara drove the ball across goal for Veysel Sari to slot home in the 51st minute.
The Turkish side soon took full control of the second qualifying-round contest to deprive Saints of a glamour tie against Marseille.
St Johnstone could not match the visitors in possession in their first European tie for 13 years but Tade's running did cause the visitors problems in the first half.
Tade missed the first leg with a hamstring injury and came in along with fellow summer signing Hasselbaink. Sean Higgins and Jamie Adams dropped to the bench as Saints started with a 4-5-1 formation.
Eskisehirspor had won the first leg by two deflected goals but were dominant in their home ground and they controlled the early stages as Saints struggled to keep the ball.
The Turks created the first chance when former Brazil international Dede picked out his fellow full-back Sari with a cross. The right-back got in front of Hasselbaink and dived to head but Alan Mannus got down well to block.
The Perth side created their first opportunity when Hasselbaink robbed Sari after a throw and cut the ball back for Chris Millar 22 yards out, but the midfielder sliced his first-time shot out for a throw-in.
Tade had a better chance moments later when he ran on to Callum Davidson's long ball down the left channel but he lacked conviction and flicked his effort wide.
The same two players combined for the opener in the 35th minute. Davidson won the ball in a challenge on the left wing as Saints pressed their opponents following a throw and curled an inch-perfect cross over the head of goalkeeper Ruud Buffin to allow the unmarked Tade to nod into an empty net from three yards.
Hasselbaink joined Tade up front for the second half against a powerful but not particularly mobile central defence and almost levelled the tie in the opening stages. Liam Craig won possession on the left wing and sent a low cross into the Dutchman's stride 12 yards out but Buffin palmed his shot away from the near post.
At this point, there was real belief among the 6,023 McDiarmid Park crowd that St Johnstone could pull off a shock but the visitors punctured that in the 51st minute.
Kamara was given too much room to drive the ball across a crowded six-yard box and the ball fell for Sari beyond the far post as the right-back slotted home.
Steve Lomas's side now needed three more goals and Craig passed up an early opportunity to get one after being played into yards of space by Hasselbaink. However, Buffin came out quickly and blocked the midfielder's shot.
The Belgian goalkeeper soon got down to save Millar's drive but the visitors then took all sting out of the game. Tade ran out of steam and was replaced by Sean Higgins but Saints struggled to get forward and both sets of fans took to affectionately mimicking each other's songs amid a lack of goalmouth action.
There was a late flurry of action though. Buffin tipped over Craig's free-kick and Mannus pushed Kamara's long-range effort wide before Eskisehirspor substitute Atdhe Nuhiu missed a sitter with the last kick of the game.