Sevilla, who trailed 2-1 from the opening match, had looked to be heading out of the competition when Diego Costa and Radamel Falcao gave their opponents the lead.
Jesus Navas gave the home side hope before the interval, but a red card for Gary Medel in the second half complicated matters further.
Ivan Rakitic's eventual equaliser came too late for Sevilla, who also saw Geoffrey Kondogbia sent off in the dying seconds to cap a miserable evening
"To reach a final is very important for Atletico," Falcao said afterwards.
"We went in search of a result from the beginning. We knew they would come at us and leave spaces. We made the most of it and took our chances.
Costa, who had scored both Atletico goals from the penalty spot in the first leg, put the visitors into a sixth-minute lead after carving an opening for himself before firing low into the far corner.
The Brazilian striker then broke away down the left and crossed low for Colombia's Falcao, who darted in front of his marker to score on the half-hour mark.
It had taken Sevilla 20 minutes to create their first meaningful chance, but they finally dragged a goal back with six minutes of the first half left, Navas smashing home an angled shot off the right post following a cross from Jose Antonio Reyes.
That sparked a spell of pressure from the home side and they should have been level three minutes before the interval but Rakitic failed to finish Medel's assist and directed a header wide from an unmarked position.
Sevilla pushed again in the second half but Alvaro Negredo was denied after 53 minutes by a parried save from visiting goalkeeper Thibaut Courtois.
Moments later Costa should have put the match beyond doubt for Atletico but he failed to make the most of his chance from a few yards out and, under pressure from a defender, poked a shot wide of the left post.
Medel saw red after picking up two yellow cards in the same incident - firstly for dissent, then for appearing to kick Costa while he was on the ground.
Atletico failed to finish off their opponents though, and Sevilla grabbed a late if immaterial equaliser in stoppage time when Kondogbia's through ball found Rakitic, who finished right footed into the bottom corner.
The match then finished on a sour note for Kondogbia, who saw a straight red in the final seconds for bringing his foot down on Costa.
"We know Costa is provocative but we fell for it," Negredo told Spanish television after the game.
Atletico will now face Real Madrid in the final on May 18.
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