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The Catalan giants have now won just one of their last six in all competitions and have seen their lead at the top of La Liga completely cut by Atletico Madrid and Real Madrid in recent weeks.
Atleti's win over Granada earlier in the day ensures that they are now level with Barca at the summit, behind their opponents by the virtue of a weaker head-to-head record this term, while Real are one point further back.
A bad month for Barcelona was made even worse when conceding twice in the first half and, despite Lionel Messi bagging a milestone goal to pull one back, it proved to be yet another night of frustration.
It would have been an entirely different story in Catalonia had the hosts' famous attacking trio not each squandered glorious chances to open the scoring in the opening quarter of the match.
A quick free kick moments later caught Los Che a little off guard, but Neymar was this time the man to be thwarted when his lobbed attempt was acrobatically punched clear by Alves.
Valencia found themselves very much holding on at this stage, but Rodrigo had a big opening of his own to make the breakthrough after being smartly played through, yet he rather rushed his shot and failed to even find the target from the edge of the opposition box.
It was turning into a frustrating affair for Barca when Suarez nodded over from Javier Mascherano's flick-on just a couple of yards out, and soon after the visitors were ahead when Guilherme Siqueira's cross from the left was turned into his own net by Ivan Rakitic.
Gomes again had to be alert to deny Messi 40 minutes in, as the out-of-form forward glanced a header towards goal having initially been denied from a free kick 22 yards out.
That miss again proved costly thanks to Santi Mina's well-taken goal on the brink of half time, rounding off a patient attacking move which culminated in Daniel Parejo playing the ball through the middle for his teammate to tuck home.
Luis Enrique sent his underperforming stars out a few minutes early prior to the restart, and it almost had the desired effect when Rakitic sent a shot narrowly wide when picked out by Suarez.
Valencia had conceded at least once in each of their last 22 league games heading into this one, but the Catalan giants were unable to create any other clear-cut chances against the strugglers prior to the hour mark on another evening to forget.
They did finally manage to get on the scoresheet 64 minutes in, though, when a simple move ended with Jordi Alba sliding the ball into a central position for Messi to bag his 500th senior goal for club and country.
That strike also ended a barren run of five games without a goal for the Ballon d'Or holder, providing Barca with a real lifeline for the remainder which they struggled to really capitalise on.
Rakitic had a big opportunity to make amends for his first-half own goal when found inside the box with the ball at his feet, although an inspired Alves was there once more down to his right to push away the resulting shot.
Neymar and Gerard Pique both saw shots blocked inside the box in a frantic final 10 minutes, while the Spanish defender also dragged an effort wide when in on goal at the death.
That proved to be it in terms of the reigning champions' hopes, meaning just a second home defeat of the campaign - coming fresh on the back of losing to Real Madrid here - and now plenty to do if they are to retain their league crown in the remaining weeks of an intriguing season.