The home side should have taken the lead with only four minutes on the clock. Gareth Bale was released through on goal, but after advancing into the penalty area, he curled his shot wide.
After that, Madrid continued to have the majority of the play, but they failed to fashion another clear opening.
With 20 minutes gone, Carlo Ancelotti's men had had a huge 65% of the possession, while Galatasaray's one attempt had flown high over the bar.
Felipe Melo gave away a free kick, and although it was a long way out, Bale's effort swerved and dipped into the corner of the net.
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Bale then missed a good chance to restore the lead, firing over the bar after latching on to a pass.