Jose Maria Gimenez opened the scoring in spectacular fashion just past the quarter-hour mark with his first goal for the club, and a Mario Mandukic strike just after the break was enough to seal the win for Diego Simeone’s men.
Victory takes Atletico to within a point of leaders Real Madrid, who are in action later at home to Celta Vigo, while Barcelona host Espanyol on Sunday.
In what was a tense first half of few chances, Los Rojiblancos scored with their first attack of note, and as so often happens, their goal came from a dead ball.
After Guilherme Siqueira had seen a cross headed behind, the Brazilian combined with Koke and Arda Turan from the resulting corner, feeding Uruguayan defender Gimenez, who instantly controlled the ball before firing into the roof of the net from the edge of the box to open his account for the club.
From then on, Atletico dominated possession, but they were thankful to goalkeeper Miguel Angel Moya, who managed to tip over a cross-cum-shot from Faycal Fajr five minutes before the break.
Eight minutes after the restart captain Gabi won possession just inside the Elche half and clipped a delightful ball over the top for Mario Mandzukic, whose near post effort surprised Przemyslaw Tyton in the home side’s goal.
Antoine Griezmann replaced Mandzukic with 17 minutes to go as Simeone looked ahead to Tuesday’s Champions League visit to Juventus, and the Frenchman saw an effort smothered by Tyton as Atleti continued to dominate.
However, despite failing to add to their tally, the visitors comfortably held on to secure their fifth consecutive win in all competitions.