Substitute Adrian Ramos added a fourth late on as the visitors eased to all three points in Istanbul.
Below, Sports Mole takes a closer look at the key talking points.
On target: 1
On target: 10
Was the result fair?
There is no doubt that Dortmund deserved their victory tonight. Galatasaray needed to produce a big performance, and quite simply, they just did not turn up. It was easy from start to finish for Jurgen Klopp's side as they took advantage of the home side's defensive frailties. Galatasaray saw plenty of the ball, but as the statistics suggest, the difference between the two sides in the final third of the pitch was huge. In the end, the margin of victory could have been even bigger.
Cesare Prandelli won't have any positives to take from such a damaging loss. Galatasaray's hopes of reaching the next round look to be gone, and they can only blame themselves. They failed to close down the likes of Reus, Henrikh Mkhitaryan and Shinji Kagawa, who were all gifted the space to have an impact on the game. In possession, it was an awful night for the hosts as they repeatedly gave the ball away just as it looked like they would cause problems for the Dortmund defence. Goran Pandev and Burak Yilmaz barely had a kick of the ball, while Wesley Sneijder was just a passenger on a disappointing evening for Prandelli's side.
Borussia Dortmund's performance
Dortmund continue to struggle in the Bundesliga, but they were at their best once again in the Champions League as they tore through the home side time and time again. Aubameyang was a constant threat as he linked up with the attacking trio behind him, while the defence managed to deal with any rare attack from Galatasaray. Klopp's side displayed a ruthlessness in the final third of the pitch as they attacked at pace whenever possible. If Dortmund can replicate this form in the league, they will climb the table very quickly.
Sports Mole's man of the match
Marco Reus: Mkhitaryan or Aubameyang could have collected this award on another night, but it was the class of Reus which really stood out in Istanbul. He used the ball brilliantly and troubled the hosts with his clever runs from wider positions. The German capped off a superb display with a stunning 25-yard strike, which even invited applause from the home supporters.
Aubameyang displayed a calm head in front of goal to score twice from close range early on, but he somehow managed to miss the opportunity to complete a first-half hat-trick when he struck the post from just a few yards out.
Antonio Mateu Lahoz avoided making any mistakes on a pretty easy night at the office, and he spotted a dive from Sneijder in the first half when the hosts were appealing for a penalty.
Galatasaray: Prandelli's side return to action against Istanbul Basaksehir on Sunday evening.
Borussia Dortmund: Klopp will be hoping his side can produce a similar performance when they host Hannover on Saturday afternoon.