Slovenian midfielder Amedej Vetrih scored the opener for Gaziantep in the 10th minute. Vetrih found the net with a close-range finish.
Fenerbahce equalized in the 24th minute as Greek midfielder Dimitrios Pelkas' close-range flick beat Gaziantep goalkeeper Gunay Guvenc.
Gaziantep led the game again near the end of the first half as Romanian midfielder Alexandru Maxim successfully converted a penalty.
The home team led the first half 2-1.
Gaziantep clinched the 3-1 win in the injury time of the second half as Kenan Ozer scored a classy free kick. Tonight means Fenerbahce has a two-game losing streak in the league. Last weekend the Yellow Canaries lost to Yeni Malatyaspor 3-0 in Istanbul.
In the next fixture, Fenerbahce will face defending Super Lig winners Medipol Basaksehir at home.
With 23 points in 13 weeks, Fenerbahce are in the number three spot.
Fourth-place Gaziantep boosted their points to 21 to chase Fenerbahce in the standings.
Gaziantep have one match remaining as well.
Aytemiz Alanyaspor lead the Super Lig with 26 points, followed by Galatasaray with 23. Both Alanyaspor and Galatasaray have played 12 matches so far.
Saturday's results:
Goztepe - Atakas Hatayspor: 0-1
Yeni Malatyaspor - Kasimpasa: 2-0
Trabzonspor - Caykur Rizespor: 2-1
Gaziantep FK - Fenerbahce: 3-1
