The visitor wasted plenty of opportunities in last week's goalless first leg and fell behind in the tie when Marcao headed Galatasaray ahead after 28 minutes on Friday (AEDT).
However, Pedri fired Barca back on level terms before half-time and Pierre-Emerick Aubameyang continued his good form by putting his side ahead for the first time.
Xavi's side saw out the victory at a hostile NEF Stadyumu to make it 11 games unbeaten in all competitions and take its place in Saturday's draw for the next round.
Jordi Alba blocked a goal-bound Bafetimbi Gomis shot and Frenkie de Jong shot wide from a one-on-one as both sides pushed for an early opener.
The breakthrough in the tie arrived just before the half-hour mark when home skipper Marcao got in front of Ferran Torres to guide in Alexandru Cicaldau's corner.
But that lead lasted for all of nine minutes as Pedri was played in by De Jong, skipped past two players and coolly tucked past Barcelona loanee Inaki Pena.
Aubameyang then sent a looping header against the top of the bar but tucked into an empty net four minutes into the second half after Sergio Busquets' initial shot was saved.
Galatasaray could not muster any further clear-cut opportunities and the frustration of home fans told late on when missiles were thrown onto the field.
Alba kicked the ball into the stands in response and was issued a yellow card but he was smiling minutes later when, after Torres wasted a chance to add a third when firing against the bar, the full-time whistle sounded.