Mert Hakan Yandas opened the scoring for Fenerbahce before Diego Angelo equalized for the home team.
Diego Perotti converted a penalty to give Fenerbahce a 2-1 lead going into the break at Ankara's Eryaman Stadium.
The second half was a one-sided affair as goals from Perotti, Jose Sosa, and Ozan Tufan took the visitors to a comfortable victory.
The win sees Fenerbahce leading the table with 20 points, while Genclerbirligi find themselves in the relegation zone with 5 points.
In other Super Lig matches, Gaziantep FK defeated Yukatel Denizlispor 1-0 in an away game, while Fatih Karagumruk drew 1-1 against Sivasspor at home.
