The Lions' Turkish wing forward Kerem Akturkoglu scored the winning goal in the eighth minute at Altay's Alsancak Mustafa Denizli Stadium.
The 23-year-old fired a left-footed strike to score from a narrow angle.
In the second half, Altay took risks and found a couple of chances, but Galatasaray goalkeeper Fernando Muslera made critical saves to secure the away win.
Galatasaray have 47 points in 34 matches to move to 11th place in the league table.
Altay from Turkiye's west are in 17th spot with 30 points, 10 from safety.
The 2021-22 Spor Toto Super Lig season will conclude in four weeks.
