Ozil started and finished the scoring at the Bernabeu as Jose Mourinho's side clinched a victory which ensured they ended the weekend still 13 points behind leaders Barcelona, but which more importantly will send them into Wednesday's Champions League semi-final first leg against Borussia Dortmund in good spirits.
"We know the kind of player he is - what he creates, even when he isn't as brilliant as we are used to," Karanka said of Ozil on www.realmadrid.com.
"He didn't play in Bilbao last week because he was carrying a niggle and to have these sensations after playing for 90 minutes was important for him. He played well.
"As the Champions League match is on Wednesday, he has an extra day to recover. To have these feelings and gain that confidence after the match is good for him, especially because of the great performance he produced."
Ozil broke the deadlock at the Bernabeu on the stroke of half-time and then wrapped up the scoring with his second right at the end.
In between those two strikes, Karim Benzema had put Madrid 2-0 up before Betis pulled a goal back through Jorge Molina's 73rd-minute penalty.
Marcelo and Luka Modric had to be substituted with physical problems, and Karanka acknowledged the manner of the Croatia midfielder's departure was worrying.
"Marcelo has felt it quite a lot," he said. "Luka looks like fatigue, but he said he felt something and we'll need to see how it develops. It seems to be something to do with the muscle. When we asked him, he said he felt something and that is the worst news from the match."
The match was also notable for the full debut of 21-year-old Brazilian midfielder Casemiro, whose display impressed Karanka.
"A few weeks ago they asked me about him," the Madrid number two said of the on-loan Sao Paulo man.
"We knew about him. He is a player that has played several games in Brazil and is a world champion with the Under-20s. In a match at home, with injuries in the team, we needed players with personality and experience and honestly he played well. It was good for both him and (defender) Nacho."
On 23-year-old Nacho, a product of Madrid's youth academy, Karanka added: "He put in a great performance. He's been our player since the start of the season.
"When we need him he always performs. He is the Castilla (reserve team) captain and one of the leaders. Furthermore, he trains with us and makes progress and plays in the matches that he can for us. It's important for him to combine the two things."