The hosts looked to be coasting to three La Liga points when they went 4-0 up at the Vicente Calderon thanks to goals from Mario Suarez, Koke, Arda Turan and a Falcao penalty.
But Simeone then "rushed" into making three changes - bringing on Cristian Rodriguez, Tiago and Adrian in place of Diego Costa, Suarez and Arda - and was left to sweat at the end as three goals in the last eight minutes from substitute Andrija Delibasic (two) and Leo got Rayo right back in contention.
Atletico held on, but Simeone accepted after the game that his hastiness was to blame for the close call.
"I think the team played a good game, given they had to play against such courageous opponents," the Argentinian said.
"I think they were doing pretty well. Then I rushed into the changes. The buck stops with me for all three goals we conceded.
"I hurried the substitutions and the responsibility for the end of the game is mine."
He continued: "The positives were the first 70 minutes, which was very important for the whole team - above all because the players understood the way of playing and carried it out."
The result maintained Atletico's unbeaten start to the season, lifting them to fifth in the standings - with a game in hand on the top four.
Simeone refused to set targets for what his team could achieve this term, though.
He said: "Where Atletico Madrid needs to look is at the internal commitment, the strength of the squad, for each player to understand that you can give nothing away, that nobody gives you anything and that we will mark our own destiny day by day."