Los Rojiblancos had gone in front when Diego Godin headed in, but a late Sergio Ramos header forced extra time before Gareth Bale, Marcelo and Cristiano Ronaldo scored the goals that secured victory for Real.
Simeone told reporters: "You have to look at it overall. Madrid were better in the second half, they kept us in our half and we couldn't get out. Football is wonderful because of that. People say that winning is the most important thing, but the support we've had from people tells you there's another side to it. I told my players that, when you've played as well as they did tonight, to keep their heads up and start thinking about next season.
"You can lose or you can win - this time it's our turn to lose and we know we've given it our all. People responded to what the team have done and that allows us to grow further and continue to compete.
"It's not sadness, it's bitterness. I feel bitter that I didn't reach our objective. I wish I could have won the way I wanted, but I couldn't - though I can overcome this. I know that, once you've given your all, there's other players and another match. In life, and in football, you've got everything one day, then you've got nothing the next - you have to keep going."
Atletico beat Real and Barcelona to the La Liga title.