Valencia's setback meant leaders Barcelona and Real Madrid pulled further away at the top following victories at Sporting Gijon and Deportivo Coruna on Saturday.
Undefeated Barca have 52 points from 20 matches, Real 47, Valencia 39 and Sevilla 36.
Spain striker Negredo broke the deadlock in a scrappy match at the Sanchez Pizjuan in the 21st minute.
France defender Julien Escude leaped to flick the ball on from a corner and Negredo turned the ball past goalkeeper Cesar Sanchez from close range.
Valencia's pacey Spain forwards David Villa, David Silva and Juan Mata battled to haul their side back into the match but the visitors fell further behind in the 69th.
Sevilla winger Jesus Navas burst clear through the centre and released Negredo, who lifted an exquisite chip over Sanchez from the edge of the penalty area.
Villa thumped the ball against a post and fluffed another chance before defender David Navarro netted a header in added time.
DEFENSIVE LAPSES
Atletico Madrid's miserably erratic season took another turn for the worse on Sunday when they lost 2-0 at home to Malaga and were whistled off.
Atletico are through to the King's Cup semi-finals but another league defeat left them mired in 13th place in the standings with only six wins from 20 matches.
They fell behind in the third minute at the Calderon when unmarked midfielder Duda drove a Jesus Gamez cross past goalkeeper David De Gea from close range.
With Atletico playing their Cup semi-final first leg against Racing Santander on Thursday, coach Quique Sanchez Flores left Sergio Aguero and Jose Antonio Reyes out of his starting lineup and the home side struggled to get back into the match.
They looked more of a threat with the pair on the pitch after the break but more woeful defending allowed Javi Lopez to sweep home Malaga's second on 70 minutes.
"We have to ask for forgiveness from the fans," Sanchez Flores told a news conference.
"I'm not going to look for excuses," he added. "The match was generally very bad and the result and our impotence hurt."
Villarreal lost 2-0 at home to Osasuna, a defeat that cost coach Ernesto Valverde his job.
Bottom club Xerez secured only their second win of the campaign, coming from a goal down at home to beat Real Mallorca 2-1, but remained six points adrift of fellow promoted sides Tenerife and Real Zaragoza.
Mallorca slipped to fifth, level on 34 points with Depor in sixth, while seventh-placed Getafe were held to a 0-0 draw at home by Racing Santander, the Madrid club's first draw of the season.
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