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Los Blancos created plenty of early opportunities and Angel di Maria's cross caused keeper Roberto to charge out of his goal, but when he failed to deal with the delivery, Sergio Ramos's header wasn't directed towards goal.
Carlo Ancelotti's side continued to enjoy the better chances and went close again midway through the first half when Marcelo's accurate cross was met by Karim Benzema and the French international's volley was only inches past the post.
Real thought that they had taken the lead minutes later when Benzema's point-blank effort ended up in the back in the net, but the former Lyon striker was adjudged to be offside by the assistant referee.
Granada had tried to be adventurous during the opening period, but the side from the Spanish capital continued to enjoy the best goalmouth opportunities, and when Cristiano Ronaldo's acrobatic effort seemed destined for the back of the net, it took the reflexes of Roberto to keep the ball out.
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Although Real failed to find a way through before the break, they dominated possession and territory in the minutes after the interval.
Early in the second half, Benzema again threatened the opposition defence and latched onto Luka Modric's cross, only the see his effort come back off the frame of the goal.
Moments later, the hosts finally found a goal and it came the reliable boot from Ronaldo, whose effort from the edge of the box found the bottom corner of Roberto's net.
After falling behind, the visitors' defending became more desperate as Real enjoyed plenty of possession in the final third of the pitch and came close to a second goal when Ronaldo's thumping free kick came back off the bar.
Real did increase their advantage 16 minutes from time when Marcelo's driving run into the box and cross reached Benzema, who fired home from close range.
With six minutes remaining, Di Maria almost added a third goal when his shot from the edge of the box trickled narrowly wide, but the hosts had done enough to seal the three points.
The result means that Real go top of La Liga for the first time since they won the league title in 2012, although they have played a game more than both Barcelona and Atletico Madrid.