Star striker Costa lasted only nine minutes of Atletico's 4-1 loss to Real Madrid in Lisbon before making way, battling the hamstring injury he had in the lead-up to the decider.
Simeone accepted responsibility for playing Costa, who he said appeared in worse condition than on Friday.
"It was my responsibility to have (Costa) play and obviously I made a mistake because I had to switch him as early as I did," he told a news conference.
"Obviously he wasn't as good as he had been the day before. That was my decision to make.
"We looked at each other, we caught each other's eye, and we didn't want to waste part of the game with one less player."
Diego Godin's first-half header had La Liga champions Atletico on track, but Sergio Ramos levelled in the 93rd minute.
Simeone's men looked tired in extra-time and Real capitalised through goals from Gareth Bale, Marcelo and Cristiano Ronaldo.
But there was no sadness for Simeone.
"It's not sadness, it's bitterness. I feel bitter that I didn't reach our objective. I wish I could have won the way I wanted, but I couldn't – though I can overcome this," he said.
"I know that, once you've given your all, there's other players and another match. In life, and in football, you've got everything one day, then you've got nothing the next – you have to keep going."
Simeone has already urged his players to start looking ahead to next season while praising Real for their efforts in the second half.
"You have to look at it overall – Madrid were better in the second half, they kept us in our half and we couldn't get out," he said.
"Football is wonderful because of that. People say that winning is the most important thing, but the support we've had from people tells you there's another side to it.
"I told my players that, when you've played as well as they did tonight, to keep their heads up and start thinking about next season."
