Croatian midfielder Ivan Rakitic scored the decisive goal on 70 minutes yesterday to leave Valencia, who hit the woodwork twice, empty-handed as Sevilla did their European hopes the power of good.
Valencia surrender third place to Villarreal, who won 1-0 at Athletic Bilbao on Sunday, while Sevilla move level on points with sixth-placed Bilbao.
Valencia had lost 3-1 to Schalke in the last 16 of the Champions League followed by a 4-0 mauling at Real Zaragoza in their last league outing and their slide continued with an unfortunate loss to Sevilla.
Sevilla had the first real opening on the quarter hour mark with Jesus Navas forcing Valencia goalkeeper Vicente Guaita into a smart save.
Valencia heeded the warning and upped their game with Robero Soldado going close a minute later and Valencia were furious to see penalty appeals turned down for a foul on Joaquin.
On 23 minutes Spain international forward Alvaro Negredo crashed a shot onto the post as the visitors almost made the breakthrough.
Valencia hit the woodwork themselves in the second half with Brazilian striker Jonas Goncalves smashing a shot against the right hand post from an acute angle.
Valencia then hit the crossbar on 59 minutes with Juan Mata's chip tipped onto the bar by Sevilla goalkeeper Javi Varas.
Varas then denied Jonas with his chest minutes later as Valencia pushed for the opening goal.
Sevilla hit Valencia with the killer blow at the other end with Rakitic producing a brilliant first time volley with his left foot on 70 minutes.
"Three points here in Valencia are great for our European chances," said Varas. "It was an attractive game tonight and both teams had their chances. We are glad to come away with a hard earned win."
Villarreal benefited from Valencia's slip with a 1-0 win over Athletic Bilbao at San Mames lifting them into third on goal difference.
A Marco Ruben header on 58 minutes clinched all three points for Villarreal who host Barcelona in their next league match.
"It is an important step towards the Champions League," said Villarreal coach Juan Carlos Garrido. "There are nine games left and finishing third or fourth will depend on if we are capable of playing like we did tonight."
Villarreal looked jaded after their Europa League commitments -- they reached the quarter-finals by beating Bayer Leverkusen 2-1 on Thursday -- but improved after the interval and took the lead just before the hour.
Striker Ruben, deputising for the injured Nilmar, headed in a fine Borja Valero cross to silence the home fans.
Bilbao could have equalised on 69 minutes but substitute Markel Susaeta shot straight at Diego Lopez.
Former Real Madrid goalkeeper Lopez was also called into action four minutes from time to thwart a fine header from Llorente.
Earlier, Venezuelan striker Jose Rondon took his league tally to 12 goals with a vital brace as Malaga moved off the bottom of the table with a 2-0 home win over European hopefuls Espanyol.
Rondon struck on seven and 25 minutes as Malaga won two consecutive league games for the first time this season.
It allowed them to climb off bottom spot and they are now just one point from safety.
For Espanyol it was a seventh league defeat in nine games as they cling on to fifth place.
Promoted Hercules, the only side to beat Barcelona in the league this season, replace Malaga at the bottom following a 4-0 home loss at the hands of Osasuna.
Barcelona continue to lead the way in Spain and stayed five points ahead of rivals Real Madrid with a 2-1 home win over Getafe on Saturday.
Real responded with a nervy 2-1 victory over derby foes Atletico Madrid at the Vicente Calderon later the same day.
Real were indebted to goalkeeper Iker Casillas, who pulled off a string of saves to deny Atletico, who have not beaten their neighbours since 1999.