The away side took a one-goal lead into the interval after Odion Ighalo's header found the back of the net on the 30-minute mark.
Alcorcon improved in the second period, but they conceded again late on when Ighalo completed his brace with a smart finish from inside the box.
Read how the first leg unfolded in Sports Mole's minute-by-minute live commentary of the action below.
ALCORCON: Gimenez; Serran, Verdes, Mora, Juli; Christian, Martinez, Iribas, Prendes; Jony, Angel
GRANADA: Karnezis; Nyom, Mainz, Murillo, Angulo; Iturra, Rico, Yebda; Brahimi, Ighalo, Piti