Emre Kilinc gave Galatasaray a 1-0 lead at the New Hatay Stadium in Antakya, Turkiye.
The Lions failed to double the lead when Patrick van Aanholt missed the penalty.
Before the first half ended, Kerem Akturkoglu scored a remarkable free-kick to put Galatasaray ahead again.
Hatayspor took the control from the beginning of the second half and Mohammed Kamara found the net to level the score.
Galatasaray were reduced to 10 men when Taylan Antalyali was given a straight red card for headbutting Ruben Riberio in the 82nd minute.
The home side completed the comeback victory after Saba Lobzhanidz scored twice on the counterattacks.
While it was Galatasaray manager Domenec Torrent's first game as in charge, Hatayspor jumped to the third place with 35 points.