Cristian Herrera's 82nd-minute effort had looked set to earn the struggling hosts a crucial three points, but their hopes of moving four points clear of the bottom three were dashed.
Instead they gained a two-point cushion over the drop zone, the draw no more than they deserved from a match in which they frustrated European hopefuls Sevilla for long periods.
The visitors' unbeaten run in the league stretches back to early November, but two dropped points on Sunday saw them miss the chance to move above Villarreal and into seventh, at least temporarily.
Neither side offered much in a turgid first half short on chances.
Carles Gil fired over from way out for Elche, while Carlos Bacca shot wide from much closer range at the other end.
Both teams livened up after the break, with Elche giving as good as they got.
Javi Marquez shot wide from 25 yards before Richmond Boakye's header from Edu Albacar's free-kick was tipped round a post by Antonio Beto.
Bacca then had two chances to give Sevilla the lead, but first shot weakly wide when well-placed inside the area and then saw an effort blocked by Ruben Perez del Marmol.
But Elche went in front in the 82nd minute when Herrera headed Fidel's left-wing cross into the bottom corner.
The hosts could not hold out, though, as Carrico snatched a point for Sevilla by heading home a Jose Reyes corner.