The Brazilian slotted home twice in the second half to claim a slender advantage in this Copa del Rey semi, as the hosts finished with 10 men and the visitors with nine
All three goals came from the spot, after the referee had adjudged players from both sides to have handled in the area. Alvaro Negredo replied for the visitors, but Emir Spahic and Fernando Navarro were both dismissed either side of Diego Godin seeing red for the hosts.
Atleti were still without the injured Radamel Falcao and opted for Adrian Lopez to partner Costa up front, whilst the visitors opted for Negredo in a lone role.
Both sides were limited to very few chances in the opening stages of the match, Costa having the first shot off the game, but his effort was always going wide of Beto's goal.
Next to attack was Arda Turan, his free-kick sailing over the crossbar minutes later as the hosts began to get a hold of the game, but Sevilla were defending well.
Half an hour in, and the first real chance of the game came through Koke, but after brilliant work by Adrian, the midfielder could only shoot straight at Beto.
Negredo's shot from distance was acrobatically tipped over by Thibaut Courtois as the visitors ended the half on a high.
The opening goal came on the 48th minute through Costa's penalty. Spahic handled the ball in the area and was given his marching orders for a second yellow, allowing the Brazilian to put his side ahead from the spot.
But seven minutes later the sides were level through a Negredo penalty. It was a harsh decision as the ball bounced up onto Diego Godin's arm whilst preventing a goalbound effort and he was also given a straight red for his troubles.
Ivan Rakitic's long range effort went narrowly wide of Courtois' goal as the tempo of the game increased after the break.
Sevilla were starting to gain control of the game for the first time in the match, and were unlucky to go behind.
A third penalty was awarded after Fernando Navarro was adjudged to have handled in the area with 20 minutes remaining, and Costa duly cooly despatched for his second of the night.
The game ebbed and flowed as the visitors looked for an equaliser, but persistent fouling from Sevilla saw Atletico awarded numerous free-kicks that started to kill the game off.
Navarro was shown a second yellow for bringing down Costa with four minutes remaining and one last chance came from Mario Suarez but he headed wide from a Turan cross, meaning Atleti hold a slight advantage ahead of the second leg in a month's time.