Simeone's side were within seconds of lifting the trophy for the first time following Diego Godin's 36th-minute opener.
But Sergio Ramos headed home an equaliser in injury time and Real pushed on in extra-time to win 4-1 thanks to goals from Gareth Bale and substitute Marcelo and a penalty from Cristiano Ronaldo.
In ugly scenes, Simeone lost his head in the closing seconds and ventured onto the pitch to confront Real's Raphael Varane before being sent from the field as the final whistle blew.
Despite the defeat, Simeone said he was proud of his side, who had claimed La Liga glory last weekend for the first time since 1996.
He said: "It's a painful defeat when you give 100%. Now we have to rest for next season. I'm proud of my players.
"I told my player that it's pointless to cry. They must be proud of themselves and think of next season.
"I don't feel sadness, just bitterness that I didn't achieve objectives. In life, one day you have everything, the next, nothing.
"Our fans should be proud of our excellent season and not waste a second being sad. I give my players 9 1/2 out of 10."
Atletico's gamble on the fitness of star striker Diego Costa proved flawed with only nine minutes gone when the Spain international was forced off by the hamstring injury he sustained just a week ago against Barcelona.
Simeone added: "It was my responsibility and obviously I made a mistake."