Atletico broke the deadlock four minutes into the second half after Emir Spahic had handled in the penalty area.
Negredo did the honours from 12 yards, but Sevilla's parity lasted just 15 minutes until referee Miguel Ayza Gamez pointed to the spot for a third time.
Fernando Navarro nudged the ball with his hand after falling over in the 18-yard box and Costa doubled his tally with minimum fuss.
Navarro was not sent off for the incident that led to the final spot kick, but he was eventually dismissed five minutes from time following a professional foul on Costa.
The second leg will take place at Estadio Ramon Sanchez Pizjuan on February 27.