The Gabonese striker tapped in his first after six minutes, following a perfect pass from Marco Reus, and made it 2-0 from close range when Lukasz Piszczek crossed from the right and found him unmarked.
Reus made it 3-0 just before the break with a brilliant long-range lob past keeper Fernando Muslera.
Substitute Adrian Ramos completed the rout with almost embarrassing ease in the 83rd, only seconds after he had come on, with an assist from Ilkay Gundogan.
Galatasaray had more possession than their German opponents, but only managed to find the target once from nine efforts on goal.
The result preserved Dortmund's 100 percent winning record in Group D -- with nine points from three games -- and kept them three points ahead of second-placed Arsenal, who snatched a last-gasp 2-1 win over Anderlecht. Galatasaray and Anderlecht both have one point each.
"The match was over after 30 minutes, after that we let them run after us," Dortmund defender Neven Subotic said.
Dortmund midfielder Gundogan added: "We only gave Galatasaray one or two chances. I think we had a very good game tonight and we could have scored even more."
Galatasaray midfielder Hamit Altintop, who spent eight years playing in the Bundesliga, said: "I think we need to criticise ourselves if we are running 10 kilometres less than our opponents during a game at this level.
"We need to ask ourselves if we are doing the right moves on the pitch. You only achieve success if you really make an effort and struggle for it."
Galatasaray coach Cesare Prandelli said his team would turn their focus on winning the domestic league.
"We need to improve and become a better team, we want a championship in the league. Our target has always been to win the trophy at home," Prandelli said.
(Editing by Alan Baldwin/Stephen Wood/Pritha Sarkar)
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