Diego Martinez's side had won its past two league games and led in the second minute, Ramon Azeez heading in at the back post, and it made it 2-0 from the penalty spot.
Arturo Vidal was penalised for handball after a VAR (video assistant referee) check and Alvaro Vadillo sent Marc-Andre ter Stegen the wrong way to leave Barca, which brought Lionel Messi and Ansu Fati on at half-time, with a mountain to climb.
Barca is winless in seven away games in all competitions and struggling in seventh on the table, while promoted Granada is the shock league leader after a third straight clean sheet.
Junior Firpo was making his first LaLiga start for Barcelona but he made two errors as Granada took the lead with 63 seconds on the clock.
Antonio Puertas profited on Junior's twin mistakes to race clear and deliver a low cross that ballooned over Ter Stegen via a deflection off Gerard Pique for Azeez to nod home.
Luis Suarez hit a free-kick into the wall as Barca tried to strike back, but it ended the first half without recording a shot on target.
After Messi came on at half-time, Barca almost gifted Granada a second. Ter Stegen carelessly dropped a cross, the goalkeeper recovering in time to beat Roberto Soldado to the loose ball.
Granada doubled its lead in the 66th minute, Arturo Vidal conceding a penalty shortly after replacing Ivan Rakitic, with Vadillo beating Ter Stegen from the spot.
Despite Messi and Fati adding zest to their attack, Barca still could not find a way through, the captain forcing Rui Silva into his first save with eight minutes to play.
Barca has now taken a single point from three away league games this season, while it drew a blank at Borussia Dortmund in the UEFA Champions League in midweek.
Messi's lack of fitness has exposed the weaknesses in Barca's squad and questions will continue to be asked of Valverde.
Granada's remarkable start to the season means it tops the league ahead of Sevilla, which has a game against Real Madrid in hand, on goal difference, but Barca is only three points off the summit despite its issues both at the back and in attack.
Barca has a quick turnaround with Villarreal visiting Camp Nou on Wednesday (AEST). Granada goes to Real Valladolid on the same day.