Atletico defeated Inter Milan in a penalty shootout after the aggregate score finished at 2-2.
Simeone said: "I have a great admiration for our people the welcome they gave us today when we arrived at the stadium and at the start of the match was very emotional.
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"We came into this from a bad game against Cdiz and, honestly, what I was trying to convey to my players was that this was not a time to ask (for the fans' support) but to give.
"Our fans are decisive in our stadium. The whole world could see that, and it made our players grow today against an incredible opponent. Koke gave his all out there, and Griezmann went on until he couldn't run any more.
"Memphis (Depay) came on and showed up when we really needed him. This is what we needed. Being among the last eight might sound easy, but it isn't."