The Catalans came into the round of 16 second leg level on aggregate after a 0-0 draw in the first leg at the Camp Nou on March 10, before sealing passage with the win in Istanbul.
The hosts took the lead shortly before the half-hour through Marcao, but Pedri finished off a nice team move by Barca with a lovely display of individual skill to draw his team level before the break.
Pierre-Emerick Aubameyang put Barca ahead three minutes into the second half with a close-range header after two magnificent stops by Galatasaray goalkeeper Inaki Pena, who is on loan from the Spanish club.
Ronald Araujo came on for United States international Sergino Dest, who limped off in the 56th minute with an apparent hamstring injury just one week before the Americans play Mexico in the first of their final three CONCACAF World Cup qualifiers.
Barca ran down the clock as Galatasaray pushed for an equaliser, infuriating the home crowd who started throwing bottles and plastic cups onto the field. The referee stopped the game for a few minutes while the Galatasaray players tried to calm their fans down.
Despite the sideshow, Barca stood tall and saw out the result to book a spot in the quarterfinals, with the draw set to take place on Friday.