Daniel Parejo’s deflected free kick in the 41st minute opened the scoring for Valencia, who had not won in La Liga since March 2, but Kevin Gameiro levelled in the 86th.
Negredo, however, restored the home side’s advantage a minute into stoppage time.
Valencia moved up provisionally to 13th in La Liga, nine points clear of the relegation zone, after picking up their first win in five league matches. Sevilla, the only team in the league without an away win, stay seventh on 48 points.
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Valencia dominated the first half and after spurning seven chances and having a penalty appeal turned down, they went ahead when Parejo’s free kick from 30 metres deflected off the Sevilla wall and flew into the top corner.
The visiting side improved in the second half and Fernando Llorente, Vitolo and Coke missed chances to level before top scorer Gameiro struck Michael Krohn-Dehli’s threaded pass beyond Valencia keeper Diego Alves.
Yet former Valencia coach Unai Emery was denied a happy return to the Mestalla when Negredo pounced on a loose ball in the area to score just before full time.
Celta Vigo boosted their hopes of qualifying for European competition by beating Sporting Gijon 1-0 thanks to a solitary strike in the 64th minute by Spain international Nolito.
Celta tightened their grip on fifth by moving on to 52 points, while Sporting are 18th on 28, two short of Granada above them.
(Editing by Toby Davis)
