Former Liverpool striker Dirk Kuyt scored the only goal of the game for Ersun Yanal's men, who held on to take all three points, with Ozgur Ileri sent-off for the hosts.
Galatasaray remain mired in mid-table, and new manager Roberto Mancini will be hoping to gain momentum quickly after the defending champions limped to a 1-1 draw with second-placed Rizespor on Saturday.
Engin Baytar opened the scoring for the hosts at the Turk Telecom Stadium, but striker Tevfik Kose secured a draw for Rizespor, before Selcuk Inan was dismissed for the hosts.
Besiktas suffered their first defeat of the season as they were beaten 2-0 by Antalyaspor on Monday, Sengalese attacker Lamine Diarra scoring a brace to ensure the points for the home side.
Trabzonspor moved into the top four with a 2-0 home triumph over Konyaspor, defender Mustafa Yumlu netting twice to seal a third straight win for Mustafa Recit Akcay's team.
Sivasspor emerged victorious in arguably the most entertaining game of the weekend, as they won 4-2 at Elazigsporon Sunday.
Goals from Manuel da Costa and John Utaka put the visitors in command, before a strike from Ahmet Gork and a Tidiane Sane penalty levelled things up.
Defender Da Costa was then sent-of for Sivasspor, but the 10 men rallied to take the points, striker Aatif Chahechouhe finding the net twice, before Caglar Birinci was dismissed in the closing stages for Elazigspor.
Gaziantepspor claimed their first win of the season as they overcame Karabukspor 3-0, Cenk Tosun, Serdar Ozbayraktar and Artem Milevskiy scoring in a superb victory.
Elsewhere, Eskisehirspor won 2-0 at Kasimpasa, Kayseri Erciyesspor and Kayserispor played out a 1-1 draw, while Bursapor's game with Akhisar Belediyespor ended goalless.
