The hosts made a perfect start to the contest when Ivan Rakitic's free kick was nodded into the top right corner by Negredo after just four minutes.
They almost doubled this lead less than 10 minutes later when Jose Antonio Reyes found space to shoot inside the box, but he couldn't steer his effort on target as it flew wide of the left post.
Sevilla came within inches of making a similarly fast start to the second half when Coke's close-range effort came back off the post in the 51st minute.
Beto was called into action again 10 minutes later, getting down well to keep Ander's low effort out.
Sevilla's chances to regain the lead took a major blow in the 73rd minute when they were reduced to 10 men courtesy of a second yellow card for Fernando Navarro.
However, they were only at a numerical disadvantage for seven minutes as Aymeric Laporte was also given his marching orders for a second yellow 10 minutes from time.
Both sides looked to find a winner as the match became an end-to-end spectacle, but it was Sevilla who got it in the 87th minute. Reyes received the ball on a quick break before crossing it to Negredo, who made no mistake with his finish.