The LaLiga giants had to fight from a goal down to take all three points.
Aleksey Batrakov opened the scoring for Galatasaray, before Barca replied with goals from Joao Cancelo, Fermin Lopez, Anthony Gordon and Karim Adeyemi.
Bayern Munich drop down to fourth position, despite their come-from-behind victory over Viking.
More matches will be played across Europe on Wednesday night.
