Slaven Bilic's side had led Turkey's top flight for five weeks thanks to six straight wins, but they relinquished first place to Fenerbahce after their third loss of the campaign.
Following a goalless first half Felipe Melo fired Galatasaray in front after 50 minutes, with Burak Yilmaz securing the win in stoppage time.
The result moves Gala level with Besiktas on 35 points, with leaders Fener just one point clear after their 2-0 victory over Istanbul Basaksehir on Saturday.
Goals from Dirk Kuyt and Alper Potuk lifted Ismail Kartal's side to the summit, while Basaksehir lost further ground in fourth place.
Bursaspor moved to within one point of them with a comfortable 3-1 with over Akhisar Belediyespor, while Kasimpasa are three back after edging a seven-goal thriller 5-2 against Kayseri Erciyesspor.
Meanwhile, Mersin Idmanyurdu held Genclerbirligi 1-1 in the race for a top six spot, but at the bottom Balikesirspor were denied a second successive win as Oscar Cardozo rescued a 2-2 draw for Trabzonspor.
Two goals from Ronald Vargas in three first-half minutes looked to have given Balikesirspor a memorable victory, but Mehmet Ekici and Cardozo – who scored in stoppage time – ensured a share of the points
That late disappointment leaves them three points adrift at the bottom after Sivasspor beat Karabukspor 2-0, while Rizespor earned a 1-1 draw against Konyaspor.
Finally, Ertugrul Saglam's Eskisehirspor sit a point above the bottom four after a 3-1 defeat to Gaziantepspor.
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