Atletico took an early lead in the match through Celta defender Carl Starfelt's own goal, controlling the match until Clement Lenglet was dismissed after picking up his second yellow card in the 40th minute.
The home side dominated the second half and eventually got the equalizer in the 68th minute after Celta's veteran Iago Aspas tapped in following a good save from Jan Oblak.
Despite dropping two points, Argentine manager Diego Simeone showered his side with praise for their gritty performance.
"11 against 11, we had it under control," said Simeone after the match.
"With 10 men, we behaved well defensively; we couldn't do so in attack, except for one from Koke. I'm very happy with the team's work. I'm keeping the positives. The team responded well in a complex situation, and they responded well.
"I left very happy. Very happy. Celebrating the work of the lads."
Atleti head into the international break in fifth position on the LaLiga table and tenth on the UEFA Champions League table.