Cristiano Ronaldo opened the scoring early in the first half before Karim Benzema doubled the hosts’ lead.
Substitute Gonzalo Higuain’s second-half header put the result beyond doubt, and made Madrid’s task in the second leg at the Turk Telecom Arena next Tuesday look far easier.
Galatasaray’s coach and players had insisted before the game that they would try to attack the Primera Division champions and would not settle for holding out for a draw.
They were loyal to their promise in the first half, but it left them very exposed at the back.
Madrid had their first attempt on two minutes, Benzema heading just over the bar after a cross from Angel Di Maria, and they took the lead after just nine minutes.
Xabi Alonso supplied the ball up to Mesut Ozil from his own half and the German played a one-two with Benzema, which allowed him space to deliver an excellent low pass into the path of Ronaldo, who calmly chipped the ball over goalkeeper Fernando Muslera and into the net.
Gala played pretty football but were limited to shots from outside the area.
Didier Drogba turned Raphael Varane on the 18-yard line and shot just over the bar, while Wesley Sneijder also missed the target from 20 yards.
Ozil could have doubled the lead when a long Sergio Ramos header out of defence led to Madrid attacking, but the midfielder, receiving the ball from Ronaldo, could not put his shot past Muslera.
Madrid’s second goal arrived just before the half-hour mark. Michael Essien delivered a high ball into the box which fell at the feet of Benzema, who was in space, took one touch to control the pass and slotted home from six yards.
Galatasaray’s best chance of the half fell to Emmanuel Eboue just before the interval.
The former Arsenal player burst through on goal but hit his shot straight at Diego Lopez’s legs.
Sneijder, unimpressive in the first 45 minutes against his former club, was removed at half-time and replaced by defender Gokhan Zan — a move which saw Fatih Terim’s side switch to a back three, with Eboue pushing up to right midfield.
The tactical change did not help the visitors’ attacking ambitions, and they went on to pose even less of a threat after the break as they had before it.
Madrid also dropped their intensity in the second period. Ronaldo and Di Maria both had chances to test Muslera, the better of which fell to the Argentinian, but the goalkeeper had little trouble with either effort.
Higuain came on for Benzema on 65 minutes and seven minutes later made the triumph secure when he headed in a free-kick from Xabi Alonso.
Jose Mourinho’s side rode out the final minutes of the game with little trouble. Alonso and Ramos both picked up yellow cards which mean they will miss the second leg.
Achieving victory was much lighter work for Madrid than their two hard-fought games with Manchester United in the previous round but, barring a complete collapse in Istanbul, they can be confident of reaching their third consecutive Champions League semi-final under Mourinho.
REAL MADRID: Diego Lopez, Essien, Sergio Ramos, Varane, Fabio Coentrao, Alonso, Khedira, Di Maria (Pepe 86), Ozil (Modric 80), Ronaldo, Benzema (Higuain 65).
GALATASARAY: Muslera, Eboue, Kaya, Nounkeu, Riera (Amrabat 83), Altintop (Bulut 78), Felipe Melo, Sneijder (Zan 46), Inan, Burak Yilmaz, Drogba.
Referee: Svein Oddvar Moen (Haugesund).