Jurgen Klopp's men produced a trademark display of powerful counter-attacking football and the game was essentially out of sight by half-time as Dortmund raced into a 3-0 lead at the Turk Telekom Arena.
Pierre-Emerick Aubameyang opened the scoring when he applied the finishing touch to Marco Reus' excellent low delivery and the Gabon international doubled his tally shortly before the 20-minute mark.
Reus then put Dortmund in firm control in the 41st minute with a stunning long-range effort.
There was a sour note for Dortmund when Sven Bender was forced off with an apparent elbow injury in the second half, but Adrian Ramos' goal a minute after coming as a substitute completed the rout.
The victory was a welcome relief for Dortmund who suffered their fifth defeat in eight Bundesliga matches this season at Cologne on Saturday.
However, Dortmund continue to excel in Europe and their third pool victory means another win in the return match with Galatasaray on November 4 will confirm their spot in the knockout stages.
It is a completely different story for the hosts, though, who have just one point and face an uphill task to progress.
Dortmund were ahead after just six minutes when Henrikh Mkhitaryan set Reus free down the left and his excellent centre across the penalty area was side-footed home at the back post by Aubameyang.
Galatasaray pushed forward in search of an equaliser, with Alex Telles a threat on the left, but the visitors were a constant threat on the counter-attack and the tactic yielded a second goal in the 18th minute.
Mkhitaryan released Lukasz Piszczek beyond left-back Telles on the right and the Poland defender picked out the unmarked Aubameyang, who adjusted his feet well to sweep beyond Fernando Muslera from just outside the six-yard box.
Dortmund were finding gaps in the home defence with ease and Shinji Kagawa should have done better when he shot tamely at Muslera when one-on-one, while Aubameyang wasted a chance for his hat-trick when he inexplicably hit the post with the goal gaping from Mkhitaryan's delivery.
However, the third arrived four minutes before half-time when Reus let fly with an unstoppable 25-yard shot that crashed over Muslera's despairing dive and in.
Dortmund had to replace Bender with Matthias Ginter in the 55th minute, but it failed to disrupt the away side's flow and Reus should have hit the target when his half-volley deep inside the penalty area flew high into the stands.
The Germany international wasted another chance when he shot straight at Muslera from the edge of the box, before Kagawa dragged wide from 20 yards after showing neat foot work.
But Ramos capped the result in the 83rd minute as he collected fellow substitute Ilkay Gundogan's throughball and kept his cool to give Dortmund a huge shot in the arm ahead of a home game against Hannover when they return to domestic action this weekend.
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