After last week's semi-final, first leg at the Vicente Calderon in Madrid ended 0-0, Atletico fell behind to a strike from former captain Fernando Torres in the 36th minute of the return encounter at Stamford Bridge.
However, they refused to throw in the towel and goals from Adrian, Diego Costa and Arda Turan drove them into the final of Europe's elite club competition for the first time since 1974, setting up a meeting with wealthy city rivals Real Madrid.
"I want to thank the mothers of these players we have at Atletico Madrid because they gave birth to them with balls this big," an emotional Simeone, holding out his hands as if he were grasping an imaginary football ball, told a news conference.
"The reaction of the team will stick in people's memories," added the Argentine, who was known as a never-say-die player during his career at Atletico and various clubs in Spain and Italy.
"The players took control of the pitch against a powerful team, with a lot of history in the Champions League, and that make me very pleased.
"We have to be grateful for their effort because without them we could not have experienced what we are."
Real, the record winners with nine continental titles, will be playing their 13th final in Lisbon on May 24, while Atletico have featured only once, a reverse to Bayern Munich forty years ago.
They will become the first two teams from the same city to contest any European final.
"Everyone here in Madrid is happy," Real coach Carlo Ancelotti told Sky Italian after Atletico joined his side in the final.
"Those who have experienced these things know that there are no favourites in a final," added the Italian, in his first season in charge at Real. "It is going to be a very balanced match."
LEVANTE FOCUS
After failing to beat their rivals for more than a decade, Atletico have enjoyed some welcome success in recent meetings with Real, beating them 2-1 after extra time at Real's Santiago Bernabeu stadium to win last season's King's Cup.
They followed that up with a 1-0 victory at Real in La Liga in September before last month's 2-2 draw at the Calderon.
However, Real knocked them out 5-0 on aggregate in the last four of this term's King's Cup.
Simeone warned his players there was little time to celebrate the victory over Chelsea as they needed to recover for Sunday's La Liga match at Levante.
Atletico remain on course for a double of Champions League and Spanish league titles and are four points clear of second-placed Barcelona, where they play on the final day of the season, with three games left.
Real, who beat Barca to win the King's Cup last month and can still claim a rare treble, are two points behind Barca in third with a game in hand.
"Tomorrow we have to focus on the Levante game which is very important and not on any other goal," Simeone said.
(Editing by John O'Brien)
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