Diego Godin's first-half goal looked to have given Los Rojiblancos enough for the first European Cup success, but a late Sergio Ramos header forced extra time before goals from Gareth Bale, Marcelo and Cristiano Ronaldo secured victory for Real.
He was quoted by Sky Sports News as saying: "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 players 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."
Simeone was sent to the stands just before the end of the game, after he ran onto the pitch to confront Raphael Varane in the wake of Ronaldo's late goal for Real.