Granada scored once in either half against the Spanish champions, who have now lost two of their five league matches at the start of the 2019-20 campaign to drop to seventh in the table.
The victory is El Grana's first over the Catalan giants since April 2014, leaving Barca boss Ernesto Valverde with much to ponder amid a difficult start to the season.
Granada - promoted from the Segunda Division last term - incredibly took the lead in just the second minute of the match when Antonio Puertas's deflected cross was headed home from close range by Azeez, who simply could not miss.
Gerard Pique attempted to block the delivery from Puertas, but the ball looped towards the far post where Azeez was on hand to send the home side into a shock lead.
Barca had a lot of the ball in the spell that followed, but Granada had another opportunity to score a second in the 14th minute when Puertas struck one from the edge of the box, only to see the ball deflect just wide.
The away side's frustrations were shown by the fact that Luis Suarez picked up a yellow card in the 21st minute of the match for dissent, with the Uruguay international certain that his side should have had a spot kick.
Granada continued to cause huge problems on the break, and Darwin Machis saw a dangerous low cross cleared by Pique in the 26th minute following another quick counter-attack.
Suarez had a half-chance for the visitors in the 32nd minute after capitalising on a loose piece of defending, but the Uruguay international saw his shot blocked before Sergi Roberto was dispossessed shortly after.
Carlos Neva then fired over the Barca goal in the 41st minute with Granada looking the more likely to score a second in the closing stages of the first half at the Estadio Nuevo Los Carmenes.
A late scramble saw the visitors have a couple of half-chances to level the scores through Pique and Suarez - both booked in the first period - but the referee's whistle was blown with Granada leading 1-0 at the interval.
That said, the home side almost bizarrely scored a second in the 52nd minute when Ter Stegen spilled a routine cross, but the Germany international did manage to respond just in time to prevent a goal.
Roberto Soldado left the field to a standing ovation from the Granada supporters just before the hour, and Azeez's early goal still separated the two teams heading into the final 30 minutes of action.
Another scramble inside the Barca box saw Granada have a chance in the 64th minute, but Ter Stegen just about managed to gather after a host of bodies attempted to make contact.
VAR had spotted something in the melee, though, and Arturo Vidal's handball was punished, handing Granada a huge chance to double their lead from the penalty spot through Vadillo.
The 25-year-old had only been on the field for four minutes but found the back of the net with a cool spot kick to spark jubilant scenes inside the stadium.
Fati looked to halve the deficit with a low strike in the 68th minute, but Granada goalkeeper Rui Silva was on hand to make a smart save as Barca's frustrations continued.
Ter Stegen then had to keep out a strike from Puertas down the other end with the home side threatening to wrap up the points with a third goal, before Silva prevented a low Messi effort from finding the back of the net in the 83rd minute.
The Spanish champions continued to push in the latter stages of the contest, but it was a terribly disappointing night for Valverde's side, who needed more from the likes of Suarez and a seriously off-colour Antoine Griezmann.
Next up for Barca is a home game against Villarreal on Tuesday, while Granada - now first in the league - will attempt to make it four straight La Liga wins when they travel to Real Valladolid on the same night.
GRANADA (4-3-3): Silva; Diaz, Duarte, Sanchez, Neva; Herrera (Gonalons 83'), Azeez, Montoro; Machis (Vadillo 62'), Soldado (Fernandes 58'), Puertas
BARCELONA (4-3-3): Ter Stegen; Semedo, Lenglet, Pique, Junior (Ansu 46'); Roberto, De Jong, Rakitic (Vidal 62'); Griezmann, L Suarez, Perez (Messi 46')