Having had a realistic chance at a treble just a few weeks ago, Barca finish the season with only the league title to show for their efforts, a situation likely to increase Valverde's unpopularity among supporters.
Barca were way off their best in the first half at the Benito Villamarin and deservedly found themselves trailing 2-0 at the break, as Kevin Gameiro and Rodrigo Moreno clinically punished the defending champions.
Valverde's men pushed forward desperately after the break and Lionel Messi pulled one back shortly after being denied by the post, but their dominance proved futile as Valencia held on to win the Copa for the first time since 2008.
The warning signs were there for Barca after just five minutes, as Gameiro pressured Clement Lenglet into a wayward pass at the back and Rodrigo pounced, skipping past Jasper Cillessen and seeing his effort blocked on the line by Gerard Pique.
Neither Cillessen nor Pique could spare Barca 16 minutes later, however, as Jose Gaya charged forward on the break and picked out Gameiro, who shook off Jordi Alba and blasted home.
Rodrigo doubled the lead just past the half-hour mark, heading in Carlos Soler's inch-perfect cross from close range.
Malcom and Arturo Vidal were introduced for Nelson Semedo and Arthur at half-time, as Barca pushed for a comeback.
Messi remained the one to watch, though, hitting the left-hand post with a gorgeous outside-of-the-boot effort just before the hour after a one-two with Malcom.
But he was not to be denied in the 73rd minute, as Messi tapped in after Lenglet's header came back off the upright.
Valencia squandered two chances to finish Barca off on the break deep into stoppage time, as Goncalo Guedes put wide when one-on-one and also missed an open goal from distance, but they held on to consign Barca to a disappointing end of the season.
What does it mean? Valverde's future surely in doubt
If Valverde's popularity was not already a matter for debate, it certainly is now. With a LaLiga and Copa double, you could justifiably see him hanging on to the job for another season, but now things look a little cloudier, particularly given the increasing fan criticism of his leadership.
Arguably the only thing saving him at the moment is Messi's apparent backing.
Rodrigo and Gameiro lead by example
They got the goals, but Rodrigo and Gameiro did so much more as well. The pair led the line brilliantly, never giving Barca's defence a moment's rest, while their link-up play – with each other and the rest of the team – was exceptional.
Clement's long day
It started poorly for Lenglet, as the Frenchman made an awful error after just five minutes from which Valencia should have scored. He rarely looked comfortable thereafter, with Gameiro and Rodrigo giving him the runaround.
Given they finished the season with just one trophy, a big close-season awaits Barca. There are doubts about Valverde's suitability, while their squad is short in a number of areas, such as centre-forward and full-back. As for Valencia, if they can keep hold of their star players and bolster their squad for the Champions League, they will be confident of another promising campaign next term.