With the hosts desperately seeking to end a two-match losing run, Suarez unleashed a stunning drive from distance to open the scoring in the first half, before Garry Rodrigues capped off the win with a fine second goal late on.
Rayo Vallecano, who were reduced to 10 men when midfielder Raul Baena picked up two yellows towards the end, are now only three points clear of 17th-placed Elche after losing their third successive match.
The hosts, looking for only their fifth win this season on Monday night, made most of the early running but found chances hard to come by.
Cristian Jonathas was guilty of squandering the best of them when he could only head over from 10 yards in the 14th minute, despite finding himself in space and under little pressure.
Given their rock-bottom status going into the match, missing such opportunities could have created uncertainty in Elche’s ranks, but Fran Escriba’s side stuck to their task well and their relentless approach soon paid off.
Los Franjiverdes finally got the goal their dominance deserved when right back Suarez unleashed a rocket from 30 yards in the 21st minute that gave Rayo keeper Antonio Tono absolutely no chance.
The 26-year-old Uruguayan pounced on a loose clearance after a scramble in the Rayo penalty area and, with the visitors slow to push out, he put his foot through it first time and watched it fly into the top right corner.
Visibly buoyed by Suarez’s first goal of the season, Elche penned Rayo back and David Lomban and Jonathas both went close to doubling their lead with headed opportunities – the latter forcing a superb, and unorthodox, save from Tono, who flicked it clear with his right leg.
Rayo finally showed signs of life before the break without really creating anything of note and they had to be relieved to be only a goal down at the interval.
The visitors tried to force the issue early in the second half and they almost got their reward when the ball broke for Roberto Trashorras just inside the box but he showed poor composure and blazed a rushed shot wide.
Chelsea loanee Mario Pasalic nearly gave Elche a two-goal cushion in the 63rd minute, but he failed to get his glancing header on target after a rapid Elche counter.
Rayo, who are now only four points clear of the drop zone, went down to 10 men when Baena picked up two needless yellow cards in quick succession to be sent off with 15 minutes to go.
And Elche made their man advantage count as substitute Rodrigues cut in from the right, rode three tackles, and played a deft one-two with midfielder Faycal Fajr before stroking past Tono to rubber-stamp the hosts’ return to winning ways in the 82nd minute.