Imanol Alguacil's Sociedad took an early lead when Swedish striker Alexander Isak set up Alexander Sorloth in the seventh minute. Isak got on the scoresheet himself three minutes after halftime with a clinical finish from a set piece.
However, Suarez rose to meet a pinpoint cross from Joao Felix around the hour mark to inspire the home fans inside the packed Wanda Metropolitano.
The Uruguayan completed the comeback for the champions when he was fouled in the area 13 minutes from time and once the penalty was ratified by VAR, converted it with ease.
After winning their Europa League game in Austria against Sturm Graz 1-0 on Thursday, Imanol was forced into a number of changes to his Sociedad side with players lost to injury or suspension. Right back Andoni Gorosabel was played on the left and Norwegian striker Sorloth was put up front with Isak.
That gamble paid off early on as the Swedish striker led a counter-attack from the left side and crossed to Sorloth who had the composure to put the ball to the right of Slovenian goalkeeper Jan Oblak.
Atletico were roared on by their fans but it was Isak who scored the second for the visitors shortly after halftime with a low strike that sneaked in off the left post.
Trailing 2-0, Atletico coach Diego Simeone made some attacking tweaks to his side such as bringing on Argentine winger Angel Correa. The changes had the desired effect as Joao Felix had space to set up Suarez with a perfect cross in the 61st minute and the former Barcelona striker headed in.
The move was almost replicated in the 77th minute, this time a low cross that Suarez tried to volley but was hit from behind by a defender. The referee did not initially give a penalty but, upon recommendation by VAR, went to check the pitchside monitor and changed his mind.
Suarez stepped up himself to convert the spot kick and rescue a point for Atleti, but he wasn't happy.
"The feeling is that we left two points on the pitch. It's one point that doesn't taste good," Suarez told reporters. "We played at home and we have to be stronger. LaLiga is very strong and we need to go out there with more attention to details."
"I think we are the ones that should feel a bad taste in the mouth, that left two points on the pitch," said Sociedad defender Mikel Merino. "We were loyal to our style and played in a way that I think we deserved that win."