After a 0-0 draw in the first leg of the tie last week, the Catalan giants managed to brave the intimidating atmosphere at the Nef Stadium to seal progression.
They did, however, find themselves 1-0 down just before the half hour mark when Brazilian defender Marcao Teixeira diverted a corner home from eight yards out.
Marcao headed past Marc-Andre ter Stegen to give the hosts the lead leaving the German goalkeeper with no chance.
Barcelona responded in less than 10 minutes through talented teenager Pedri, who was a constant threat for Xavi's side throughout the contest.
The Spain international collected the ball in the area from Ferran Torres before dancing and weaving his way through to leave Marcao and Taylan Antalyali on the floor, before slotting home past Barca's on loan shot-stopper Inaki Pena.
Barca then went in front four minutes after half-time through Aubameyang.
Pena had produced an incredible double save to deny both Sergio Busquets and Pedri, before Frenkie de Jong cleverly headed square in the box for the former Arsenal striker to poke home.
The 32-year-old Gabon international has now scored seven goals in his last seven games across all competitions for Barca, and has as many goals as he managed in his last 19 games for Arsenal.
There were unsavoury scenes in the closing moments, with Jordi Alba having bottles thrown at him while taking a throw-in.
The referee initially appeared set to take the players from the field, before continuing the game, after handing the Spanish full-back a yellow card for booting the ball into the crowd.
Galatasaray continued to put up a fight, but Barca always looked in control and now go into Friday's quarter-final draw.