After earning one point from its first two Champions League games, the win lifts Madrid into second place in Group A behind Paris Saint-Germain and two points clear of third-placed Club Brugge.
PSG beat Brugge 5-0 away in the other group game Tuesday.
Galatasaray nearly took the lead early on when Florin Andone got in behind Madrid’s defensive line but saw his shot saved by Thibaut Courtois.
The Belgian goalkeeper stopped another Andone effort from a corner soon after when Marcelo failed to intercept a pass.
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Madrid came into the game off a defeat to Mallorca in the Spanish league and coach Zinedine Zidane made six changes to the starting lineup. Most crucial was the introduction of Hazard, whose fine cut-back pass set up Kroos to score with a deflected shot after Galatasaray had given the ball away cheaply in defense.
Hazard could have made it 2-0 but hit the bar in Madrid’s best chance of the second half.
The loss leaves the Turkish champion last in Group A, with little hope of qualifying, and under increasing pressure as it sits sixth in its domestic league.