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 Cup 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.
"To reach a final is very important for Atletico," Falcao said. 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.