The defeat left Atletico third on 21 points from 11 games, three adrift of leaders Real Madrid and one behind champions Barcelona, who both have a game in hand.
Sociedad, who celebrated their first La Liga win against Atletico since November 2014, are sixth with 19 points from 11 matches.
The home side dominated from the start but Atletico missed the best chance in the first half when Kevin Gameiro raced clear and hit the post with a dinked shot over Sociedad keeper Geronimo Rulli.
Former Arsenal forward Vela was lively on the right flank and fired Sociedad ahead when he calmly slotted home a 54th-minute penalty after Atletico's captain Gabi tripped left back Yuri Berchiche.
Vela's jinking run produced a second spot kick in the 75th minute after he was brought down by substitute Angel Correa, with Brazilian striker Willian Jose blasting his shot past Atletico keeper Jan Oblak.
Antoine Griezmann, who scored Atletico's stoppage-time winner in a 2-1 defeat of Rostov in Tuesday's Champions League clash, saw his close-range shot cleared off the line in the closing stages as the weary-looking visitors went down tamely.
(Writing by Zoran Milosavljevic; Editing by Ian Chadband)