Atletico keeps pressure on Barcelona
Diego Simeone's side had struggled to overcome Fran Escriba's disciplined outfit until Koke broke the deadlock on 62 minutes, and Diego Costa, who had assisted the opening goal by playing to the limits of the law, made sure of the three points 11 minutes later.
Los Rojiblancos entered the game in confident mood after putting seven goals past Getafe the week before, and with Diego Costa fully recovered from a thigh strain, Simeone was able to field a full-strength side against Elche, who had beaten Valencia in their last match.
Although Atletico were in control for large parts of the game, they struggled to create many openings.
Their only chances in the first period were a shot from Arda Turan which flew just over the bar, an effort from David Villa saved by goalkeeper Manu Herrera, and a shot from Costa which had beaten the Elche stopper but was blocked on the line by Damian Suarez.
Although Elche had been more timid than their opponents, they had two decent chances before the break, Richmond Boakye shooting over the bar with just Thibaut Courtois to beat, and Ferran Coro firing narrowly wide of the far post from the edge of the area.
Fifteen minutes into the second half, Carlos Gil went for goal from outside the area, forcing Courtois to stretch across his goal to tip the ball out for a corner.
A couple of minutes later, Atleti took the lead, to the fury of the hosts.
Raul Garcia, who had come on for Villa just a moment before, made an ambitious attempt on goal from outside the area, but Manu Herrera was unable to hold on to the shot, spilling the ball at the feet of Costa.
The forward barged Alberto Botia out the way and allowed Koke to tap in from a few yards out.
The Elche players were furious that a foul had not been called against Costa, but were unable to change referee Carlos Delgado Ferreiro's mind.
Atleti had the ball in the net a few minutes later, but this time the goal was disallowed, as Arda had headed the ball out of Herrera's hands just as the goalkeeper caught a cross from Filipe Luis.
But the visitors did not have to wait much longer until they wrapped up the game.
Substitute Adrian burst forward on an Atleti counter and played a perfectly-weighted chipped pass over the heads of two Elche defenders for Costa to strike past Herrera and score his 15th goal of the season.
Atletico will now be hoping that Athletic Bilbao can do them a favour on Sunday and beat Barcelona, allowing Los Rojiblancos to share the league's summit with the Catalans for a little longer.