Diego Martinez's side took advantage of the postponed Clasico between Barcelona and Real Madrid, with Alvaro Vadillo's second-half goal sealing all three points at Estadio Nuevo Los Carmenes.
The Andalusian side have enjoyed a remarkable return to LaLiga after two seasons in the second tier, winning six of its opening 10 games, including a memorable 2-0 victory over Barca on 22 September (AEDT).
With Barca not playing this weekend after its clash with Madrid was postponed, Granada was able to move a point clear at the top of the table.
Its haul of 20 points from 10 games this season is as many as it managed from all 38 league matches when they were relegated in 2016-17.
It could be dethroned when Ernesto Valverde's side takes on Real Valladolid in its game in hand on Wednesday (AEDT) at Camp Nou.
Civil unrest in Catalonia forced the first Clasico of the 2019-2020 season to be rearranged for December 18.