Second-half goals from Miranda and Diego Costa inspired Atletico to a 2-0 victory over an Elche side that finished with 10 men at the Vicente Calderon on Friday.
The result was enough to send Atletico six points clear of cross-town rivals Real Madrid with four games remaining, though Carlo Ancelotti's men have played one game less.
"We knew it was going to be an open and dangerous match because we faced a team that works tactically well," Simeone told reporters after the match.
"We couldn't find gaps in the first half and by contrast, they took advantage of chances on the counter-attack, to create some danger.
"The attitude changed in the second half. The intensity increased and, from persistence, came the prize. Our perseverance delivered a fair victory."
Atletico's goal finally arrived with 18 minutes remaining, with defender Miranda on hand to head home substitute Jose Sosa's corner.
"I didn't like the team being so cut-off in the first half and needed time for the players to read the game on the pitch," the Argentine said.
"The changes were good. Diego gave us depth, Raul's entry was decisive. Jose, once again with his set-piece, broke up a tight match with a corner kick."
Simeone's men must now turn their attention to Tuesday's UEFA Champions League semi-final first leg against Chelsea at Stamford Bridge.
"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 said.
