The Catalans were knocked out of the Champions League in midweek and then saw their hopes of retaining La Liga rocked on Saturday evening
The Algerian's third goal of the season in the first half gave the Andalusian outfit its first win over the champions since they returned to the top flight in 2011.
Barca was dumped out of the Champions League at the quarterfinal stage by Liga title rival Atletico Madrid in midweek, and the pressure on coach Gerardo Martino was ramped up again at Estadio Nuevo Los Carmenes.
Neymar cut a frustrated figure as he spurned several chances, and defeat ensured Barca missed the chance to replace Atletico - which plays on Sunday - at the top of the table, while victory for Granada eased its relegation fears.
Martino stated in the build-up to this game that Barca's season would be deemed a failure following its Champions League exit, and he must now hope his side respond by winning the Copa del Rey final against Real Madrid on Wednesday.
Xavi, Jordi Alba and Dani Alves dropped to the bench as Martino rang the changes following the loss at Atletico and, probably, with one eye on the Copa final.
Barca made a bright start, but it was stunned when the home side took the lead just 16 minutes in.
Midfielder Fran Rico was the architect, with a pass that pierced the Barca defense. Brahimi raced away from Martin Montoya before beating Jose Pinto with a right-foot finish inside the near post.
Granada wanted a penalty when Youssef El-Arabi went down under a challenge from Javier Mascherano, but referee Carlos Delgado saw nothing untoward.
Neymar should have done better when he headed Montoya's inviting cross over the crossbar, before the Brazil forward tried his luck with a measured right-foot strike from outside the area that was palmed away by Orestis Karnezis.
Former Santos man Neymar was becoming increasingly frustrated, and he was somewhat fortunate to escape with only a yellow card when he kicked out at Allan Nyom just before halftime.
Barca got straight back on the front foot after the break and Granada was indebted to a fine save from Karnezis to keep out a Cesc Fabregas strike that looked destined for the back of the net.
Neymar was wasteful once again when he fired a left-foot strike wide of the far post 12 minutes into the second half as the home side came under yet more pressure.
Karnezis then thwarted Barca again with an outstanding reflex save from a Lionel Messi free kick, before smothering Sergio Busquets' strike from close range.
Neymar's misery continued 17 minutes from time when he volleyed wide at the far post, and Andres Iniesta then flashed a shot just wide of the far post.
El-Arabi could have sealed the points three minutes from time when he raced through on goal, but Pinto denied him.
That miss was not to prove costly, though, as the home side held on for three precious points in its battle to avoid the drop and dealt a big blow to Barca's title hopes.