Riki and Recio both had further chances for the hosts in the first half, but neither team were able to break the deadlock before half time.
However, the hosts were able to edge ahead against the run of play when Youssef El-Arabi slid the ball through for Brahimi to slot in.
Granada had a chance to double their lead less than 10 minutes later when Cristian Sapunaru handled the ball in the box, earning himself his second booking of the game and leaving Elche a man short for the final 20 minutes.
However, Fran Rico could not convert from the spot, thumping his penalty onto the right post.
Jeison Murillo came close to adding a second in the final five minutes as he tried to head in from close range, but he saw the ball saved.
The result moves Granada up to 11th in the table, while Elche sit 14th.