Lionel Messi scored the only goal of the game in the 81st minute after VAR ruled that Robin Le Normand had handled the ball inside the area. Alba had a second goal ruled out in stoppage time for offside.
The defender had celebrated by cupping his ears to his own supporters and later explained that he had been upset by whistles aimed at the players during the first half after last week's Clasico defeat to Real Madrid.
"No one likes to play poorly or how we're playing at the moment, but I don't like it when in the 15th minute with the score goalless there are whistles," he told reporters.
"No one likes it when the fans go against you, especially in your own stadium. Our attitude is good and all the players are giving everything out on the pitch. I'm not going to hide. Just as I respect everyone, they also have to respect me.
"I respect Barca fans a lot. Even falling [early] in the Champions League [in recent seasons], they have stuck with the team.
"It's not easy when things are not coming off, but we have to live with that at the end of the day. We know that if we're not at our best, it's going to be hard. We have to be strong mentally to turn the situation around."
Despite the frustration among sections of the support, Barca remain in the hunt for La Liga and the Champions League. The win over La Real took them two clear of Real Madrid -- who play Real Betis on Sunday -- and they're level with Napoli in their Champions League last 16 tie.
However, there's a strange atmosphere around the club. White hankies were waved at president Josep Maria Bartomeu on Saturday and there were calls for him to resign.
Headlines have also been made off the pitch for the wrong reasons in recent weeks, including Messi's war of words with sporting director Eric Abidal and allegations the club hired a third-party company to discredit players, club legends and potential presidents on social media.
The latest controversy came this week when coach Quique Setien's No. 2, Eder Sarabia, was criticised for his touchline demeanour during last weekend's Clasico.
Images showed Sarabia losing his cool several times, screaming a number of expletives and criticising several players, including Antoine Griezmann, for missing a first-half chance.
Alba, though, said reports the players were angered by Sarabia's behaviour -- which Setien apologised for in the week -- are nonsense and defended his persona in the dugout.
"It's as if Sarabia killed someone," the Spain international said. "Can the coaching staff not express themselves as they want? The comments he made are the most normal in the world.
"He lives the game intensely, he didn't show a lack of respect to anyone. I would have done the same in his position -- or worse. We're not annoyed, not at all."