Cristiano Ronaldo got the hosts off to the perfect start as he calmly chipped home from Mesut Ozil's wonderful through ball after just nine minutes.
Karim Benzema then doubled Madrid's lead on 29 minutes with a simple finish at the back post from Michael Essien's cross and Gonzalo Higuain headed home the third 17 minutes from time to seal a highly satisfactory evening for the hosts.
Galatasaray manager Fatih Terim had signalled his intent to attack by naming the trio of Wesley Sneijder, Didier Drogba and Burak Yilmaz in his starting line-up.
However, as a result the Turkish champions gifted way too much for Madrid to create in midfield, with Ozil in particular shining in the first-half.
The German international set-up the opener as he received Xabi Alonso's pass before turning and feeding Ronaldo to dink the ball over the advancing Fernando Muslera for his ninth Champions League goal of the campaign.
Drogba nearly produced an immediate reply for the visitors as he spun Raphael Varane on the edge of the box, but with a clear sight of goal the Ivorian lashed wildly wide of the target.
Sneijder also fired wide from 20 yards as Galatasaray went in search of the away goal.
However, that was opening up gaps that Madrid love to exploit on the counter-attack and from another sprightly break Ozil forced Muslera into a good low save.
Drogba did hit the target with another long-range strike that Diego Lopez could only turn behind for a corner and Galatasaray had a good claim for a penalty turned down when Sami Khedira appeared to handle inside his own area.
Yet, they were still struggling to cope at the back and went further behind when Essien's cross just evaded Ronaldo but fell perfectly for Benzema to slot in off the near post.
Emmanuel Eboue had a great chance to give Terim's men a lifeline just before half-time as he broke through on goal after a neat exchange of passes with Drogba, but the former Arsenal defender fired his shot too close to Lopez, who made a comfortable save.
Perhaps recognising the error of his ways to start the game, Terim replaced Sneijder with Gokhan Zan at the break.
The change to a 5-3-2 formation did allow the visitors to stem the tide of Madrid attacks in the second period, but the Spanish champions should have scored a third when Angel Di Maria was found by another majestic Ozil pass only for the Argentine to volley meekly into Muslera's arms.
The tie was virtually killed off by a third Madrid goal 17 minutes from time though as substitute Higuain glanced home Alonso's in-swinging free-kick.
And Galatasaray's night was summed up when Yilmaz saw a yellow card for diving when he was challenged by Sergio Ramos, meaning he will now miss the return-leg on Tuesday.