Former Liverpool and Barcelona forward Luis Suarez scored twice, the first in the 41st minute to put Atletico ahead after being put clear on goal by England international Kieran Trippier.
Suarez got his second after the break, shortly before Lucas Boyle scored for Elche but substitute Diego Costa sealed the win via a late penalty to make it 3-1 to Atletico.
The victory takes Atletico three points ahead of Real Madrid and Real Sociedad, while Barcelona, who drew 2-2 with Valencia later on Saturday, trail the leaders by eight points.
Suarez, making his 200th La Liga appearance, has now scored seven goals in nine league appearances for Atletico this season.
Watch the highlights of the match below
Barcelona 2-2 Valencia: Lionel Messi equals Pele’s Santos record in Barcelona draw
The Argentine, 33, achieved the feat with a diving header just before half-time, immediately after his penalty was saved by keeper Jaume Domenech.
Messi who scored his first goal for Barca in 2005 against Albacete has now gone on to score 642 more.
Pele, the Brazilian football legend, scored 643 goals in 19 seasons for Brazilian side Santos between 1956 and 1974.
Sadly for Messi, his celebrations will likely be muted after Barcelona failed to beat their mid-table opponents.
The result leaves the 26-time champions fifth in La Liga, eight points behind Atletico Madrid, who earlier defeated Elche 3-1 to move to 29 points.
Other La Liga results
SATURDAY 19TH DECEMBER 2020
- Atl Madrid 3-2 Elche
- Barcelona 2-2 Valencia
- Levante 2-1 Real Sociedad
- Osasuna 1-3 Villarreal
- Sevilla 1-1 Real Valladolid