Simeone's men have lost out in the final to cross city rivals Real Madrid in two of the past three seasons.
However, they have another chance at Champions League glory as a 0-0 draw with Bayer Leverkusen on Wednesday sealed their place in Friday's draw 4-2 on aggregate.
“Firstly there is joy,” said Simeone.
“It shows the great effort the club and, above all, the players have made in the way they have competed and given their heart and soul in the past four years.
“Everyone has done their bit to be amongst the top eight teams in Europe.”
Atletico were far from their best at the Vicente Calderon as they relied on a stunning triple save from goalkeeper Jan Oblak to prevent just a second Champions League defeat in the past four seasons.
However, the Spanish side got the job done in Germany three weeks ago thanks to goals from Saul Niguez, Antoine Griezmann, Kevin Gameiro and Fernando Torres.
“We didn't need to attack like crazy,” said Atletico's top scorer Griezmann.
“We tried to see the game out at the back and that is what we did. We are in the quarter-finals and we are all happy.”
Another European reunion with Real is one of seven possible opponents for Atletico in the last eight along with Barcelona, Bayern Munich, Borussia Dortmund, Monaco, Leicester City and Juventus.
“Now hopefully we can get a good draw and through to the semis,” added Griezmann.
“We have to show every year we are among the best teams in Europe and every year we achieve it.”
New Leverkusen boss Tayfun Korkut was taking charge of his first Champions League game and summoned a much-improved performance as they forced Atletico onto the back foot for long spells.
However, Korkut hailed Atletico's desire to defend as Oblak's miraculous triple save to deny firstly Julian Brandt and then two follow-up efforts from Kevin Volland midway through the second-half secured Los Rojiblancos' 17th clean sheet in their last 20 home Champions League games.
“I haven't seen anything like it,” said Korkut.
“I've watched it again and it isn't just him. There were many defenders throwing themselves to block the shot.
“You see how this team defends their goal. They are all there to put their foot or any part of their body on the line.”
Korkut was hired after a 6-2 thrashing at Dortmund following their first leg defeat saw Roger Schmidt sacked.
And with Leverkusen languishing in 10th in the Bundesliga, Korkut is hoping to gain momentum from their impressive showing to make a late run to qualify for Europe next season.
“We saw that we have the quality to cause them problems. Atletico suffered and we had a lot of chances,” he added.
“For me the important thing is that the team has taken the first steps forward comparing with where we have come from in the Bundesliga.”