Jose Mourinho's men had a comfortable two-goal lead at the break after goals from Cristiano Ronaldo and Karim Benzema before Galatasaray gave a better account of themselves in the second half but substitute Gonzalo Higuain wrapped up proceedings in the 73rd minute.
Madrid had the lead inside ten minutes when Mesut Ozil played a neat ball into the path of Ronaldo on the left side of the area with the Portuguese star deftly chipping over Fernando Muslera with his left foot for his ninth goal of the CL campaign.
20 minutes later and Madrid were two to the good when Benzema controlled a long ball from the right from Michael Essien with his thigh at the far post before firing home low into the bottom corner.
The Turkish side looked more at ease during the second half but did not really put much attacking pressure on Mourinho's men and it came back to bit them with just under 15 minutes remaining when Higuain, on Benzema, got free of his marker and glanced Xabi Alonso's leftwing free-kick past Muslera.
The result gives Madrid every conceivable chance of moving through to the final four, however, Galatasaray fans are sure to make the Spanish champions visit to Istanbul a very difficult one.
2012/13 UEFA Champions League quarter-final - first leg