The title-chasing Rojiblancos looked set to drop two precious points at the Vicente Calderon but late goals from Miranda and Diego Costa earned the hosts a vital victory.
The Argentine acknowledged after the game at the Vicente Calderon that he had erred in selecting Adrian on the right wing.
However, Simeone replaced the Spain international with Raul Garcia at the break before then throwing on Diego and Jose Sosa - changes that the Rojiblancos coach felt played a huge part in his troops digging out a 2-0 win courtesy of Miranda's 72nd-minute header and Diego Costa's injury-time penalty.
"We had the usual system but I decided to put Adrian on the right wing to create problems for Cristian Sapunaru, who is not left-back," the former midfield ace told reporters after the game.
"That imbalanced the team, though, and they had more people in the middle of the field when there was a second ball.
"Then, with a 4-4-2, we were stronger, more together and that made us more dangerous. Diego gave us more depth in midfield while Sosa did very well with the corner that made it 1-0."
Atletico is now four games away from winning their first Liga title since 1996 but, with just one home fixture remaining, against Malaga, Simeone is taking nothing for granted.
"Every game will be tough from here until the end," he warned. "We have three away games and that's plenty."
Simeone is simply hoping that his men can quickly recover from their ex-ertions against Elche, given next week they face Chelsea in the first leg of their Champions League semifinal tie.
"The players put in a great effort and now we have to recover as best we can to try to do well in Tuesday's game, which is very important for the club and for Atletico's history," he added.