The Liga leaders beat Chelsea 3-1 at Stamford Bridge on Wednesday to set up a tournament decider against city rivals Real Madrid in Lisbon next month thanks to goals from Adrian Lopez, Diego Costa and Arda Turan.
Simeone could not hide his delight with the Rojiblancos' remarkable response to falling behind to Fernando Torres' opener and admitted he was thrilled to have given their travelling support something special to savour.
"It was a beautiful, important moment, but what makes me happier is to reach the final for the people who paid a lot of money to come here," the Argentine coach told Canal Plus. "The reaction of the team will remain in the memories of the fans.
"I want to thank the mothers of these players for giving birth to them with balls that big. It's not only about running, but being good tactically.
"Our first goal was important and the start of the second half was very good. I don't think it's an advantage or disadvantage [to face Real]. We know them from the league, they are a powerful club and are accustomed to these situations."
Koke put in a masterful display in midfield and Thibaut Courtois pulled off a sensational save from John Terry's header in the second half, and Simeone stressed his eagerness to see both players prolong their careers at the Vicente Calderon.
"Koke has improved enormously. He's ours, he wants to be at the club and has a great future ahead of him," he said, before adding: "The directors understand it's for the best for Courtois to stay."
Jose Mourinho was seen paying tribute to Simeone's efforts in the closing minutes at Stamford Bridge, and the former midfield admitted he was proud to receive words of praise from the former Madrid boss.
"I'm grateful for his words; he's a hardworking man who has coached the best teams. Now we have to think about Levante [in La Liga this weekend]," he warned.