Costa, who had scored both Atletico goals from the penalty spot in the first leg, put the visitors in charge after carving an opening for himself in the sixth minute and firing low into the far corner.
His seventh goal in the Cup made him the competition's leading scorer this season, one ahead of Real's Cristiano Ronaldo and Sevilla striker Alvaro Negredo.
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 in the 29th.
Roared on by a noisy home crowd, Sevilla's most dangerous player Jesus Navas breathed life back in to the tie with a curling shot that went in off the far post just before the break.
Sevilla, who beat Atletico 2-0 in the 2010 Copa final, pressed hard in the second half but did not find a way past Thibaut Courtois again until Ivan Rakitic scored in time added on.
By then, Sevilla had lost their cool with Gary Medel being sent off 14 minutes from time, and Geoffrey Kondogbia was red-carded at the end, both for fouls on Costa.
Nine-times Cup winners Atletico will meet their neighbours Real in the May 18 final after Jose Mourinho's men swept past holders Barcelona on Tuesday to secure a 4-2 aggregate victory.
The venue for the match has still to be decided.
"To reach a final is very important for Atletico," Falcao told Spanish television.
"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.
On the incidents with Costa, Falcao added: "The end of the game was ugly. These things happen in games with a lot to play for, but we leave it out there on the pitch."
Costa, 24, has muscled his way into Atletico's starting team this year with some outstanding performances in recent weeks.
He was a constant thorn in the side for Sevilla with his driving runs, his power in the air, and his ability to get under the skin of his opponents.
The fiery Medel saw red after picking up two yellow cards in the same incident, the second for appearing to kick Costa while he was on the ground.
Kondogbia received a straight red after he committed a foul on the Brazilian and then stamped on him between his legs.
"We know Costa is provocative but we fell for it," Negredo told Spanish television.
Reaching the final shows the continued improvement of Atletico since Simeone took the helm just over a year ago.
The former Argentina captain, who helped the club win their last Copa as a player in 1996, guided the side to the Europa League final last season where they beat Athletic Bilbao 3-0.
They swept aside Champions League winners Chelsea 4-1 in the season-opening European Super Cup, and have surprised many by holding off Real Madrid to hang on to second place in La Liga behind leaders Barcelona for so long.
Atletico have not beaten Real in over a decade but Simeone was not prepared to discuss the final.
"It shows a lack of respect to Sevilla to talk about Real Madrid after this game against a great rival," he said.