Los Blancos went 1-0 up early on via Cristiano Ronaldo, but goals from Diego Costa and Miranda eventually helped Diego Simeone's men to their first major domestic trophy since they won the double in 1995-96, much to the satisfaction of the Colombian.
"It's an unbelievable feeling. Now we have to enjoy it. This win feels awesome," the prolific striker was quoted as saying by AS.
"We're happy with the effort we put in and that we can celebrate this win like the people wanted."
Atletico winger Arda Turan, meanwhile, was equally pleased with his side's surprising victory at the Bernabeu.
"This is the best moment in my life. It's an incredible day, probably the most important one in my life so far. We all believed in ourselves thanks to Simeone."