Real ensured Atletico's run without a win in the Madrid derby will stretch into its 14th year with a comfortable victory at the Bernabeu.
A wonderful free-kick from Cristiano Ronaldo lit up what was an otherwise disappointing first half before Mesut Ozil secured all three points with a tidy finish 25 minutes from time.
Despite the defeat, second-placed Atletico remained ahead of their neighbours in the standings but slipped six points behind Barcelona with Real a further five points behind.
But Karanka believed Real had dominated the game and were not troubled by their neighbours.
He told the Real website: "We controlled the match more because we scored first to make it 1-0. We didn't have too many problems at the back and they didn't create chances.
"At half-time we talked about playing the same way and the team started off in much the same vein, by creating chances. It was a hard-fought game, but we were in control at all times."
Karanka had expected Atletico to prove tougher opponents because of their good start to the season.
"It was a hard match in the sense that they have been playing well and with a lot of confidence. Everyone thought this was going to be the year that they could beat us. But the team has showed character.
"In the first half we controlled the game and in the second we created chances.
"Wins always strengthen the group but a team like Real Madrid is strong in character by nature.
"For us it was just another match given the points difference."