Malaga had not won on their travels for eight games previously, but Ignacio Camacho put an end to that barren run with a header in just the sixth minute.
Victory for the islanders, who were dumped out of the Copa del Rey during the week, elevates them into the ninth place ahead of the remainder of the weekend programme, and eases the pressure on manager Bernd Schuster.
Elche were reduced to 10 men on the hour mark when defender Damian Suarez picked up a second booking for a studs-up challenge on Camacho.
"I'm happy for the goal but happier for the three points that we really needed away from home," Camacho said. "We have to see the positive side to what happened in the Copa. We're out, but now we need to look to move up the league table."
Defeat for Elche was their third straight, and sees them remain in 14th.