Luis Suarez had put the visitors into a controversial lead as he appeared to be offside before slotting in at the near post, but Valencia's work rate was rewarded when Mina fired into the top corner five minutes from time yesterday.
Real Madrid are now just four points behind Barca as they bounced back from their embarrassing elimination from the Copa del Rey for fielding an ineligible player in midweek with Karim Benzema, Gareth Bale and Cristiano Ronaldo all on target in their 4-1 thrashing of Getafe.
Neville doesn't officially take charge until Sunday, leaving Valencia in the hands of caretaker boss Voro and his brother Phil Neville and the depleted hosts put up a brave fight against the superior quality of the European champions.
"There is nothing to reproach the team for," said Barca boss Luis Enrique.
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"We played a complete game at a very difficult place to come, we dominated the whole game and had by far the most chances.
Barcelona should have had the game wrapped up by half-time, but the star front three of Lionel Messi, Neymar and Suarez were uncharacteristically wasteful in front of goal.
Messi spooned the first chance over from Neymar's fine cross before the latter ballooned the ball into the stands with just Jaume Domenech to beat.
More brilliance on the left wing from Neymar teed up Messi once more, but on his weaker right foot he fired straight at Domenech.
Suarez then fluffed his lines when he tried to square for Neymar when clean through only to overhit his cross and see the ball fly to safety.
Valencia claimed Suarez was lucky to stay on the field at the start of the second period as he appeared to stamp on Aymen Abdennour's foot after a heated exchange between the two.
Yet, they made Barca pay for their profligacy when Paco
Alcacer held up a long ball over the top and laid the ball perfectly into Mina's path to slam home his first goal for the club.
At the Bernabeu, any sign of the fans' frustration at the club's latest gaffe in being eliminated from the Cup due to an administrative error was eased when Benzema fired home from Pepe's fine cross after just five minutes.
And he looked back to his best as he doubled Madrid's advantage on 16 minutes when he slotted into the bottom corner from Bale's headed cut-back after Ronaldo appeared to have been fouled inside the area.
The one-way traffic continued as Bale showed great composure to finish from Ronaldo's knockdown and score for a second consecutive league game for the first time since January.
Ronaldo finally got in on the act three minutes later at the end of a devastating counter-attack as he latched onto Toni Kroos's fine through ball to register his 17th goal of the season.
"The fact Benzema scored those two goals and contributed so much to the team going forward reinforces even more what I have said before that he is a very important player for us and makes those around him play better."
Atletico remained in front of their city rivals thanks to another solid display as they recorded their sixth clean sheet in seven games to extend their winning run in all competitions to six games.
Griezmann made the game safe in spectacular fashion 14 minutes from time as the French international blasted high into the roof of the net from Oliver Torres's clever through ball.