Moroccan forward En-Nesyri narrowly beat the offside trap to latch on to a through ball from Fernando and score the only goal of the game in the 66th minute.
Valencia were left 14th in the standings on 39 points.
Julen Lopetegui's side pulled themselves into the title race at the end of last month but their hopes were hit by a 1-0 defeat at home to Athletic Bilbao on May 3, while Eden Hazard's 94th-minute equaliser in Sunday's 2-2 draw at Real Madrid was another sucker punch.
"We're not really aiming to win La Liga, the Bilbao game left us devastated and the objective is now to get as many points as possible," he said.
"That defeat was a real shame as people wanted to see someone different win it. We then had a very difficult game against Madrid and got a point and today we were able to get back to winning."
Sevilla had the ball in the net early on through Suso but the goal was ruled out for offside. Midfielder Nemanja Gudelj later made an audacious attempt on goal from his own half but his shot just missed the target.
Valencia, who were playing their second game under caretaker coach Voro after sacking Javi Gracia, made a strong start to the second half and had two chances to go in front.
The visitors then pushed for an equaliser after En-Nesyri had broken the deadlock but Sevilla held on.