Celta were leading 2-0 when Savic was dismissed with 20 minutes remaining after being shown a second yellow card on a bad day for Diego Simeone’s side.
The hosts took the lead through Maxi Gomez before former Liverpool forward Iago Aspas doubled the lead with his first goal of the season.
Celta had a third goal overturned by the video assistant referee (VAR).
Argentine defender Gustavo Cabral was denied by a marginal offside call.
Atletico, champions in 2013-2014 and runners-up last season, have four points from three games going into the international break.
Victory propelled Celta to the top of La Liga before second-placed Real Madrid’s home match with Leganes later on Saturday.
Celta, who have never won La Liga, have seven points from three games.