The abrasive front-man led the line in the semi-final first leg against Sevilla at the Calderon, when leading scorer Falcao was out injured, and scored two penalties to help Atletico to a 2-1 victory.
In the return leg in Seville on Wednesday, he carved out the first goal for himself and set up the second for Falcao inside the opening 29 minutes, and the side held on for a 2-2 draw to progress 4-3 on aggregate.
Costa's seven goals in the cup make him the top scorer in the tournament to date.
Sevilla had four players sent off over the two legs, three of them for incidents involving the tall and powerful 24-year-old.
Defender Fernando Navarro brought down Costa when he was through on goal for a second yellow card in the first leg, and Gary Medel fouled him and then appeared to kick him on the ground 14 minutes from time on Wednesday.
Midfielder Geoffrey Kondogbia received a straight red after making a crunching tackle from behind on Costa, and then stamping on him between the legs in added time.
"These things happen in football. We know Costa is provocative and we fell for it," Sevilla captain Negredo said after the game.
Atletico coach Diego Simeone, a famously uncompromising midfielder in his day who was often described as playing with a knife between his teeth, praised his forward.
"When you lose you feel this impotence," Simeone told a news conference. "A striker with Costa's characteristics is difficult to mark, and the episodes at the end are part of the game."
Costa, no stranger to controversy on the pitch, shrugged off the incidents.
"I had it all controlled," he told reporters. "They didn't come looking for me, I just happened to be there. When I dished it out people used to call me an assassin. I know I have made mistakes, but I try to control myself more now.
"The team are playing well and this is the reason why I am doing well," he added. "I don't consider myself a leading figure and will carry on working hard to try and improve."
In the May 18 cup final, Atletico, who are second behind leaders Barcelona in La Liga, will meet city rivals Real Madrid, a side they last managed to beat in 1999.
The game will pitch Costa into another possible confrontation with Real defenders Sergio Ramos and Pepe, with whom he clashed at the Bernabeu earlier in the season.
"After the game we shook hands and there were no problems," Costa said of the last derby.
