Simeone's men lost 1-0 on aggregate to their city rivals, after going down by the same scoreline in the second leg at the Santiago Bernabeu on Wednesday.
Javier Hernandez's 88th-minute goal saw Real, who trumped Atleti in last season's Champions League final, advance without the requirement of extra-time, although Atletico were left to rue the sending off of Arda Turan 12 minutes earlier.
"I am proud of having these players. I am happy of having a team like the one we have. I completely identify with their work," Simeone told Atletico's official website.
"The club comes away stronger. We are among Europe's top eight, which is not easy.
"If the boys give their all as they did today, I tell the fans of Atletico de Madrid they should feel happy and proud of the players they have."
Simeone said the focus returns to sewing up their spot in the competition next season, with Atletico four points clear of fourth-placed Valencia in La Liga.
"The objective is the one we always had. Having won the Super Cup at the beginning of the year and close it with the third place, which is obviously what the club needs," he said.
Atletico goalkeeper Jan Oblak pulled off several quality saves before being stranded for Hernandez's goal.
"Of course we're not happy, but this is football. You can't always win," Oblak said, as reported by AS.
"We have six finals in the league. We get our heads up and we will do better in the league. We have to lift ourselves and go forward. Hopefully we improve in the league.
"This is a credit to everyone. Today we suffered. We didn't want to suffer, we wanted to score, but unfortunately, we didn't.
"Of course [Turan's dismissal] complicated life. But it happened. The weekend is another game.
"Madrid are a great team, great players. They deserved to win. We wish them luck."
