Galatasaray found an early goal in the fourth minute when Emre Kilinc scored from close range at Turk Telekom Stadium.
The Lions doubled the lead as Arda Turan scored on a quick attack in minute 12.
Goztepe cut the lead in half two minutes later on a goal by Turkish midfielder Soner Aydogdu.
Algerian midfielder Sofiane Feghouli's goal gave his team, Galatasaray, the insurance goal in minute 63.
Trabzonspor beat Hatayspor 1-0
In league's other action today, Trabzonspor shutout Atakas Hatayspor 1-0 with Brazilian defender Vitor Hugo's goal in the 74th minute.
With the victory, Galatasaray increased to 26 points in 13 games.
Alanyaspor, who are currently on top of the Super Lig with 26 points, will host Yeni Malatyaspor on Wednesday.
Third-place Fenerbahce with 23 points will face Medipol Basaksehir in a midweek match.
Besiktas, who are in fourth place with 22 points will take on MKE Ankaragucu at Ankara's Eryaman Stadium on Thursday.
Matchday 14:
Tuesday:
Atakas Hatayspor - Trabzonspor: 0-1
Galatasaray - Goztepe: 3-1
Wednesday:
Buyuksehir Belediye Erzurumspor - Yukatel Denizlispor:
Aytemiz Alanyaspor - Yeni Malatyaspor:
Kasimpasa - Gaziantep FK:
Fenerbahce - Medipol Basaksehir:
Thursday:
Demir Grup Sivasspor - Genclerbirligi:
Hes Kablo Kayserispor - Ittifak Holding Konyaspor:
MKE Ankaragucu - Besiktas:
Caykur Rizespor - Fraport TAV Antalyaspor
