The Spain striker netted in the fourth and 87th minutes to take his tally for the season to 17 in an entertaining clash where both sides finished with 10 men.
Sevilla are 10th with 41 points and eight games left to play, six short of Malaga in sixth place and the last Europa League qualification berth.
The hosts were quickly in front when Negredo flicked a header past Gorka Iraizoz from Ivan Rakitic's teasing free kick, and he was denied a second before the break thanks only to Carlos Gurpegi's outstretched boot.
Bilbao striker Fernando Llorente, the leading figure in last season's headline-grabbing Europa League and Copa del Rey campaigns, continued to show a worrying lack of confidence in front of goal.
He made way for Aritz Aduriz at halftime and Bilbao were level in the 55th when Gurpegi flicked a header from a free kick inside the far post.
Sevilla lost defender Fernando Navarro to a second yellow card in the 73rd, and Bilbao defender Aymeric Laporte followed him down the tunnel in the 80th.
With four minutes remaining, Sevilla's Jose Antonio Reyes broke down the left and his raking low cross left Negredo with a simple finish at the far post.
Marcelo Bielsa's Bilbao are 13th with 35 points, nine above the relegation places.