Dani Parejo's free-kick had given Pako Ayestaran's men a deserved half-time lead in his first home game in charge since replacing the sacked Gary Neville.
The poor defending that underminded Neville's reign cost the hosts once more five minutes from time though, when Sevilla substitute Kevin Gameiro swept home Michael Krohn-Dehli's pass to equalise.
However, Negredo pounced on a loose ball inside the area in stoppage time to the relief of 50,000 fans at the Mestalla, to move Valencia up to 13th and nine points clear of the relegation zone with six games remaining.
Sevilla are now 12 points adrift of Villarreal in fourth after the Yellow Submarine strengthened their hold on the final Champions League spot with a comfortable 2-0 win over relegation-threatened Getafe.
Denis Suarez got Villarreal off to a perfect start with the opener on 15 minutes before the on-form Cedric Bakambu sealed the points with his 20th goal of the season.
Earlier, Celta Vigo remained eight points off Villarreal in fifth as Nolito's second-half strike earned a hard-fought 1-0 win at Sporting Gijon.