The Catalans led 1-0 from the first leg at Camp Nou but Fernando Torres levelled the tie on aggregate after less than 60 seconds of play at the Vicente Calderon.
Neymar equalised only to see Raul Garcia make it 2-1 from the penalty spot. However, Barca booked their place in the last four with a Miranda own goal and another Neymar strike just before the break.
Atletico subsequently saw Gabi and Mario Suarez sent off, while Arda Turan was booked for hurling his boot apparently towards a linesman as Diego Simeone's side lost their discipline.
However, Luis Enrique did not want to be drawn on the numerous controversial incidents during the game, preferring instead to salute his team's character.
"I liked everything about my team, especially the way in which they recovered from a strange situation at the begining of the match when we went 1-0 down after a minute," the Barca boss told reporters after the game.
"Atleti are an amazing team and they are doing a great job, winning titles in a great way, while the pitch was in very bad condition. The ball was bouncing around the place like a rabbit!
"I don't know if that's down to the weather but my team adapted to the conditions and I'm very happy with the performance.
"I loved the second half. We had a lot of control."
Luis Enrique also singled two-goal hero Neymar out for special praise after an inspired performance.
"Neymar gives us a lot," the former Spain international added. "He scores, dribbles, he's brave - he played a great game."
Barca will now face Villarreal or Getafe in the semi-finals of the Copa del Rey.
