Atletico stunned Barca with two goals in the final nine minutes in a 3-2 victory on Thursday.
Alvaro Morata's penalty and Angel Correa's strike saw Atletico into the final against rivals Real Madrid and the comeback was labelled as close to impossible by head coach Simeone.
"I think we must live it with calmness, that we qualified for the final. It is important for the supporters, it is important for the club, it is important for the players," he told a news conference.
"It was a very tough game against one of the best teams in the world. They are the best when it comes to possession, and to attack with it.
"We suffered quite a lot during the first half. I think the second half was different, they kept their intensity and had chances until the 75tth minute, when something appeared that will always distinguish Atletico Madrid – the heart, the strength and the belief.
"We faced a difficult situation, almost impossible. But, we had the feeling that if we could get an equaliser, we could win the game, and that's what happened."
Atletico face another huge test in the final, taking on Real Madrid on Sunday.
Simeone paid tribute to Atletico's rivals and their boss, Zinedine Zidane, ahead of the clash.
"I don't have any doubt that Zidane is a fantastic manager, suitable for Real Madrid," he said.
"The results speak for themselves. They have changed since the beginning of the season and that's reflected on the pitch despite the absence of [Eden] Hazard, [Karim] Benzema and [Gareth] Bale.
"They have five midfielders on the pitch. They can do it if they move and pass accurately. They showed this the other day [against Valencia] with a great performance.
"We'll face a potent and strong team as it is with Real Madrid, Barcelona, Manchester City, Liverpool."