Despite Granada's strong start, it was the home side that took the lead when Nabil Ghilas picked up the ball just inside the box before lashing a low shot into the bottom corner.
Fran Rico came close to getting the sides back on level terms just before the break when he tried to curl a free kick in from just over 20 yards out, but his finish was a fraction high.
The miss was a costly one as the home side doubled their advantage in the final seconds of the half when Federico Cartabia picked out Florin Andone with a through-ball, which the striker smashed in from close range.
Piti had another chance early in the second half when he tried to drive the ball in from just outside the box, but Juan Carlos saw the finish all the way to make the save.
Granada looked increasingly desperate to force a goal as the second half progressed, resulting in a string of wasted chances, with Riki sending in a wayward cross before Alfredo Ortuno fired the ball well high of the crossbar.
The result leapfrogs Cordoba above their opponents into 18th in the table, leaving them just a point adrift of Levante.