An 81st-minute goal from Mauro Icardi - who had earlier missed a penalty - settled the tie 3-2 in the Turkish club's favour against a United side reduced to 10 men when Casemiro was sent-off late on.
It was United's sixth defeat in 10 games this season and the second in their opening two matches in the Champions League.
To make matters worse for coach Eric ten Hag, his team had twice led through goals from Rasmus Hojlund before Galatasaray struck back thanks to former United forward Wilfried Zaha and Kerem Akturkoglu.
A see-saw game in Naples ended with a 3-2 win for visiting Real Madrid.
Napoli led early through Leo Skiri Ostigard before Madrid hit back with goals from Vinicius Junior and a slaloming solo effort from 20-year-old England international Jude Bellingham.
The Italian champions levelled via a Piotr Zielinski penalty early in the second half but the visitors took the points courtesy of an own goal from keeper Alex Meret, who was unable to prevent a rebounded effort from crossing the line.
"What surprises me is that he's only 20 years old," Madrid coach Carlo Ancelotti said of Bellingham, who has been a standout for the Spanish giants this season.
"He has extraordinary qualities and he's showing how much talent he has."
Another young England star, Bukayo Saka, hobbled off as Arsenal lost 2-1 despite leading early at French club Lens, who celebrated their first home Champions League game in 21 years.
Gabriel Jesus broke the deadlock for the visitors with his 22nd Champions League goal after 14 minutes, but Lens hit back through Adrien Thomasson before Saka limped off for the third time in nine days.
Elye Wahi hit the winner with 20 minutes to go to open up Group B.
A late goal from teenager Mathys Tel earned Bayern Munich a 2-1 win - their 15th successive group-stage victory in the competition - over Copenhagen.
Lukas Lerager had given the Danish side the lead shortly after halftime, before Jamal Musiala levelled for Bayern.
Inter Milan beat Benfica 1-0 thanks to a 62nd-minute strike from Marcus Thuram.
The result lifted last season's beaten finalists level on points with Group D leaders Real Sociedad, who won 2-0 at Salzburg for their first Champions League victory in two decades.
Mikel Oyarzabal and Brais Méndez scored first-half goals for Sociedad, who are making their first appearance in nearly 10 years and last won a match in 2003/04, at Galatasaray.
A goal five minutes into added time from Jordan Teze earned PSV a 2-2 draw with Sevilla, while Andre Castro also scored late for Sporting Braga to snatch a 3-2 win at Union Berlin.
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