The 27-year-old Spaniard bagged a double in the victory on Monday, in a result that helps Sevilla's chances of playing in the Europa League next season.
The hosts went ahead in the fourth minute when Negredo rose highest to nod home Ivan Rakitic's free-kick.
Despite the early goal, both sides settled, and it was not until the 55th minute that Carlos Gurpegi added parity to the scoreline when his flicked header from Ibai Gomez's free kick deceived Beto in the Sevilla goal for the equalizer.
The visitors were handed a fresh impetus in their pursuit of a winner when Fernando Navarro was shown a second yellow card for a foul on Aritz Aduriz. Bilbao failed to capitalize on its numerical advantage, though, and it was soon 10 vs. 10 when Aymeric Laporte got himself booked twice in three minutes and followed Navarro down the tunnel.
Despite little time remaining on the clock, a late twist saw Negredo strike to claim all three points for Sevilla, tapping home from close range after Jose Antonio Reyes fired across the face of goal.
Unai Emery's men now move to within six points of seventh-placed Real Betis, who occupy the third and final Europa League place in La Liga. Sevilla has the opportunity to bridge that gap to just four points when it takes on Pepe Mel's side at the Benito Villamarin this Friday.