Yacine Brahimi's strike midway through the second half settled the contest in favour of Granada, who made it three home victories on the bounce following their successes over Villarreal and Real Betis.
It could have been a more comfortable victory for Granada in the end but Fran Rico missed from the spot in the 73rd minute after visiting defender Cristian Sapunaru had been sent off for bringing down Youssef El Arabi.
Fortunately for Rico, Granada were not made to rue that miss as they held on for a win that will further ease their relegation fears.
Granada started the weekend five points clear of the bottom three and they would have been confident of continuing their good recent home form with another triumph over an Elche side who were one place and one point below them.
However, the visitors gave as good as they got in the first half with Carles Gil and Manu Del Moral both being denied by Granada keeper Roberto Fernandez, while Javi Marquez and Carlos Sanchez Moreno saw efforts go wide.
Granada also fashioned opportunities, but Jose Recio, Ivan Riki and El Arabi could not make the most of them as the first half finished goalless.
It was a similar story after the break with both sides finding openings only for opposing keepers Roberto and Manu Herrera to produce fine stops to keep out Marquez and Recio, respectively.
Roberto's brilliant save to keep out Marquez's 25-yard free-kick in the 62nd minute was especially key, as two minutes later Granada claimed the goal that would win them the match.
El Arabi's ball in found Brahimi inside the area and the Algeria international slotted home past Herrera.
It should have been 2-0 nine minutes later but Rico fired wide from the penalty spot following Sapunaru's foul on El Arabi, an incident that earned the Elche man his second yellow card and an early bath.
The missed penalty was a let-off for Elche but last season's Segunda Division champions could not take advantage, with Moreno firing their last opportunity for an equaliser over in stoppage time.