The Croatia playmaker, a summer signing from Tottenham, took to the field for the first time at the Bernabeu when he replaced Mesut Ozil in the 83rd minute of Wednesday night's second leg as the tie finished 4-4 and Real won on away goals.
"I am very happy," Modric told AS. "It could not have started any better."
He added: "I thank the fans for the love they showed me, and my team-mates understood me very well.
"I've noticed that Spanish football is more technical than English football. I know expectations are high in this team. Now I will work very hard to help the team.
"Mourinho did not have to tell me that much. I was not really nervous - a little at first, but that disappeared when I got my first touch."
Full-back Alvaro Arbeloa saw signs of potential in his new team-mate's brief appearance, and said: "Modric did not play for long but everything he did, he did well."
Madrid's goals came from Gonzalo Higuain and Cristiano Ronaldo in the first 20 minutes and though Lionel Messi responded with a free-kick, Barca could not find the equaliser which would have won them the trophy.
Ronaldo said: "We started the match very well, we scored two goals and had a lot of confidence in our play. In the second half I think we conceded a goal just because we began to get tired. I have to improve and we need to as a team.
"This trophy will give us confidence and we have to enjoy it, but we must look forward."