Valencia finished La Liga season in fifth, and a place in the Europa League, after goals from Ever Banega and a Roberto Soldado double were not enough to secure the win they needed to claim fourth spot.
For Sevilla, the ninth-placed finish this win provides could also see them compete in the Europa League next season.
Valencia took the lead in the 12th minute when Dani Parejo set up Ever, who shot into the top corner from outside the area.
Sevilla were level with four minutes of the half remaining when Gary Medel headed the ball down and Negredo scored with an overhead kick from close range.
Negredo got his and his side's second goal from the penalty spot three minutes later after Ricardo Costa had brought down Diego Perotti.
Things got worse for Valencia in first half injury time as Jonas Oliveira was sent off for a bad foul on Alberto Moreno.
Ten minutes into the second half Valencia were back on level terms when Soldado volleyed into the roof of the goal from close range following a cross by Joao Pedro Pereira.
A minute later Sevilla were back in front as Negredo struck again from close range, this time with a low shot after being set up by Moreno.
Just past the hour Negredo got his fourth goal when he scored from close range from a pass by Ivan Rakitic.
With three minutes to go Sevilla pressed for a fifth but Jesus Navas saw his shot saved by Valencia keeper Vicente Guaita.
Two minutes from the final whistle Valencia scored their third goal through Soldado, who converted from a header by Victor Ruiz.