Atletico shipped two late goals to lose the first leg 2-1 and conceded within 24 seconds of the return before going down to a 3-1 defeat, with Simeone sent from the sidelines after Pablo Sarabia scored Sevilla's third at the Ramon Sanchez Pizjuan.
Atleti slipped 11 points behind La Liga leaders Barcelona following a 1-1 draw with Girona on Saturday and will not feature in the Champions League knockout stages having finished third in Group C behind Roma and Chelsea.
However, the Argentinian backed his team to continue fighting in La Liga and the Europa League having fallen out of two cup competitions already this season.
"The responsibility for this elimination and the Champions League is all mine," said Simeone.
"There are still four months left in the season and between La Liga and the Europa League, we'll keep competing like we always have.
"Our opponent took advantage of their chances. Sevilla finished their chances both here and in the first leg. This made them more comfortable in the tie.
"We were close to turning the match around in the first half and scoring the equaliser in the second, but [Sevilla goalkeeper Sergio] Rico played well."
Atleti's Wanda Metropolitano home will host the Copa final and Koke was disappointed they will not be there to take part.
7 - Sevilla have reached the Copa del Rey semi-finals on seven occasions since 2007; winning the trophy in both 2007 and 2010. Optimism. pic.twitter.com/slrDCEjGxq— OptaJose (@OptaJose) January 23, 2018
"We had dreams of playing the final in our stadium," Koke told beIN SPORTS .
"We're going to pick ourselves up because this team has always done so. Now we need to think about La Liga.
"I think Sevilla were better and when that's the case, you have to congratulate them.
"We need to study what we did wrong and improve from that. We have to fight to finish as high up in La Liga as possible and we also have the Europa League."
Atleti face Las Palmas in the league on Sunday.