Barcelona F.C. remain top with 54 points, but Atletico Madrid, who trail on goal difference, can leapfrog them with at least a draw at home to Real Sociedad, as can Real Madrid, who are a point behind, with a victory at Athletic Bilbao.
It appeared to be service as normal when Alexis Sanchez put Barcelona ahead early on but strikes either side of halftime by Daniel Parejo and Pablo Piatti put Valencia in the driving seat.
Lionel Messi equalised with a penalty after 54 minutes but the hosts were caught out again on the counter and Francisco Alcacer put the visitors back in front.
Barcelona continued to struggle at the back and Jordi Alba was dismissed for a second yellow card with 12 minutes left. The result left the Catalans with one win from their last four league matches.
"It's a long time since we lost here and it comes at a bad time. Valencia made the most of their chances and we couldn't do any more," midfielder Sergio Busquets told reporters.
"We started well and it continued for the first half hour with a lot of chances to score but then Valencia came into it more and we lost our way."
The victory is a boost for Valencia's European hopes after they had picked up two points from their last three matches.
"It was perfect for us to win at a place like this. It was a different Valencia performance with the desire to do well," said Parejo.
"We were ready to take on Barcelona and we knew how the game might go with them having a lot of chances and we (knew we) would have to fight for our own openings."
Before kickoff there was a minute's silence for former Spain coach Luis Aragones, who died earlier on Saturday aged 75. Aragones had coached Barca in the 1987-88 season and led them to victory in the King's Cup.
There was little to indicate early on that eighth-placed Valencia were capable of an upset, with Xavi's shot blocked by Jeremy Mathieu before Alexis scored after seven minutes with a shot from a tight angle that spun over keeper Diego Alves.
The hosts dominated with the more direct style favoured by coach Gerardo Martino, although they were leaving gaps at the back which Valencia exploited towards the end of the first half.
After visiting defender Ricardo Costa headed wide from a corner, Parejo equalised from close range a minute before the break after Sofiane Feghouli had pulled the ball back.
Valencia went ahead three minutes after the restart when Piatti headed home a loose ball inside the penalty area as Barcelona's defensive woes continued, with Gerard Pique and Javier Mascherano in particular struggling.
Messi pulled the home side level from the penalty spot in the 54th after a handball by Costa but Alcacer restored Valencia's lead just before the hour when he steered home another Feghouli assist.
Fullback Alba was sent off for a second yellow card when he attempted to prevent Piatti breaking away. Messi was then inches away from drawing Barca level near fulltime, but his shot went just past the post, consigning his side to their second league defeat of the season.
A brace from Portuguese winger Duda helped Malaga beat Sevilla 3-2 in their Andalusian derby.
Duda put Malaga ahead from the penalty spot after he himself was brought down by fullback Coke but Sevilla came back after the break.
Carlos Bacca finished clinically in front of goal and then Federico Fazio headed in a free-kick before Samuel Garcia brought Malaga level.
Duda was again on target seven minutes from the end to give Malaga a victory that gives them a four-point cushion over the relegation zone in 13th place.