Ignacio Camacho grabbed the only goal of the game at Estadio Manuel Martinez Valero, his sixth-minute header giving Bernd Schuster's men a first league victory on the road this season at the ninth attempt.
Elche finished with 10 men following the dismissal of Damian Suarez on the hour mark and have now lost three games in succession.
Full-back Suarez picked up a yellow card in the 28th minute before being booked again to leave his team, who fought gamely in search of an equaliser, with a mountain to climb.
Malaga rise to ninth as a result of their win, which leaves rock-bottom Real Betis as the only La Liga side yet to claim three points on the road.
Elche remain 14th, just four points above the relegation zone.
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