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.
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 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.
"We know Costa is provocative but we fell for it," Negredo told Spanish television.
"It shows a lack of respect to Sevilla to talk about Real Madrid after this game against a great rival," he said.
