Elche's fine run came to an end as Manuel Iturra's first-half goal guided Granada to their first victory in three matches in the Primera Division.
Iturra headed home Fran Rico's cross from the right in the 26th minute to snap a downward spiral which had seen them take five points from eight games.
Last week's 2-0 home defeat by Getafe had prompted calls by some fans for coach Lucas Alcaraz to be sacked.
However, they will be considerably happier now although for Elche the match had appeared to represent a good opportunity to claim their fourth successive victory.
Manu Del Moral headed over for Elche in the first attack of the game before Rico's 20-yard shot was blocked by Alberto Botia, the ball going out for a corner which came to nothing.
There were no openings of note before the visitors went ahead, Iturra heading home Rico's free-kick for Granada's first goal in three matches.
Del Moral shot wide before firing over shortly before half-time but the hosts had not seriously threatened the visitors' goal going into the break.
Yacine Brahimi and Jose Recio both had shots blocked for Granada with Carles Gil and Javi Marquez both missing for Elche.
Elche were not forcing anything of note and Granada almost doubled their lead midway through the second half.
Odion Ighalo's shot from the left of the area was blocked by Lomban only for Brahimi to pounce and unleash an effort which Manu Herrera pushed round the post.
Ferran Corominas fired wide for Elche in injury-time but the match ended in a defeat they could not complain about given the paucity of their attack.