Fenerbahce's Czech defender Filip Novak broke the deadlock by heading in the 35th minute in Istanbul's Ulker Stadium.
But then Serdar Aziz, another Fenerbahce defender, netted an own goal, leveling the score 1-1 in the 40th minute.
In minute 63, Genclerbirligi's Togolese forward Floyd Ayite scored the game's winning goal with a header.
With 58 points, Fenerbahce fell five points behind the Super Lig leaders, while Ankara's Genclerbirligi, who are at bottom of the standings, boosted their points to 24 on weekday 30.
Besiktas are at the top of the league with 63 points, with Galatasaray close behind in second place with 61 points.
Week 30 Fixtures & Results
Friday:
Medipol Basaksehir - Besiktas: 2-3
Saturday:
Demir Grup Sivasspor - Fatih Karagumruk: 1-0
Caykur Rizespor - Atakas Hatayspor: 1-0
Aytemiz Alanyaspor - Fraport TAV Antalyaspor: 4-0
Hes Kablo Kayserispor - Galatasaray: 0-3
Sunday:
MKE Ankaragucu - Goztepe: 3-0
Buyuksehir Belediye Erzurumspor - Trabzonspor: 0-0
Gaziantep - Yukatel Denizlispor: 2-0
Fenerbahce - Genclerbirligi: 1-2
Monday:
Kasimpasa - Ittifak Holding Konyaspor
