A fortuitous goal from Cristian Herrera after 59 minutes was all it took for Fran Escriba's side to take all three points, although they had to endure a nervy few minutes of injury time after Damian Suarez was sent off in the 90th minute for picking up two yellow cards.
The result means Almeria replace Elche as the team occupying the final spot above the drop zone, while the Alicante side move up to 15th in the table.
The hosts created considerably more chances than their opponents in the first half. Manu del Moral squandered an early opportunity to open the scoring, just failing to get on the end of a free-kick from Edu Albacar a few yards out from goal, and then Almeria goalkeeper Esteban denied Herrera, who had found himself clean through.
On the half-hour mark David Lomban put a free header just over the bar following a corner, and on the stroke of half-time Esteban again came to his team's aid, keeping out a free-kick from Albacar, while at the other end Elche goalkeeper Tono was called into action for the first time, adjusting his position to save a deflected shot from Verza.
Jonathan Zongo was unable to get past Tono from a tight angle early in the second half, while Del Moral's shot from the edge of the area went narrowly wide.
Just under a quarter of an hour into the second half, the hosts took the lead. Esteban came to punch a cross and Herrera tried to volley the rebound but slipped and completely missed the ball. It fell to Javi Marquez outside the area, who hit the ball towards goal, deflecting off the leg of Herrera on the floor and into the bottom corner.
The visitors' best chance of getting an equaliser arrived seven minutes after the goal, Helder Barbosa shooting towards goal from the left side of the penalty area following a corner, but neither Fernando Soriano nor Angel Trujillo was able to get the final touch which would have directed it in at the near post.
Suarez was shown his second yellow card in the 89th minute for a dangerous lunge, but Elche survived injury time with one fewer player to take the victory.