The Turkish outfit had recorded a hard fought 3-3 draw at the Jan Breydel Stadium a week ago in Bruges and they finished the job right in front of their euphoric home fans.
The Togo international got the party started for the hosts in the seventh minute. Ex-Manchester City teammate Gael Clichy swung in an inch perfect cross from the right and Adebayor powered a header past Ethan Horvath in the Blue-Blacks goal.
The second goal came three minutes before the half hour mark. After a long clearance that the Belgian side failed to deal with; the Togolese outmuscled his man before laying the ball off to Edin Visca at the edge of the box, who drilled a low shot that beat Horvath at his near post.
There were no further goals in the second half which meant the Turkish Super Lig outfit progress 5-3 after two legs and are just one playoff away from playing in the groupstages of Europe's elite club competition.
Abdullah Avci's side begin their domestic campaign at home against Bursaspor on August 11.