Real defeated PSG 2-1 on Tuesday night to go through 5-2 on aggregate.
Kovacic said: "It feels good. It was a difficult game. They play nice football. But we played well.
"They had the ball more than us but we were very aggressive. It's difficult to beat us when we play like that. It's amazing how it's changed when we play in the Champions League.
"This club is born to play in the Champions League."