The opening goal of the game arrived midway through the first half, though it was initially ruled out for offside.
VAR checked for offside and a foul on Savic's part but found no reason to disallow the goal, so Atletico went ahead.
illarreal almost gifted Atletico a second goal in the second half when Pau Torres was pressured into a mistake by Angel Correa. Luis Suarez's resulting shot was tipped over by Sergio Asenjo.
The local goalkeeper couldn't do much about the visitors' second goal when it did come with just over 20 minutes to go.
Joao Felix's sweet strike from the edge of the box found the bottom corner but his celebration took most of the attention. The Portuguese forward turned to the dugouts, put his finger to his lips and then shouted at length.
The hosts threatened to get back into the game in the closing stages. First, Alex Baena hit the post from the range. The 19-year-old then had a fierce shot superbly saved by Jan Oblak.
In the end, Simeone's men held their nerve to see out the rest of the match, creating that five-point gap back to Barcelona at the top of LaLiga Santander.