Atletico enjoyed an impressive start and got the opener early when Alvaro Morata scored from a rebound in the fifth minute.
The former Juventus striker had a second goal ruled out later in the half following a lengthy VAR review, due to a handball by Joao Felix in the build-up.
Atletico kept pushing but were repeatedly denied by an inspired Alex Remiro in Real Sociedad's goal.
Remiro produced a wonderful parry to push a long-range strike from Jose Maria Gimenez onto the bar just before the break.
Sadiq, making his debut at La Real after signing from Almeria on Thursday, thumped home a close-range header in the 55th minute.
Atletico's staff and players complained that the ball took a deflection off the Nigerian striker's hand before going in, but referee Cesar Grado confirmed the goal without checking the VAR footage.
"It's hard to understand the referee's criteria - or the lack of it," frustrated Atletico captain Koke told Movistar Plus.
"The referee called off our goal but didn't even go to check the image of their goal.
"It's frustrating because we had a similar issue the last game and things just don't change. We never know when the referees will or will not use the VAR."
Remiro went on to make at least three more top-class stops from close range and was largely to thank for his team's valuable point against a better Atletico side.
There was more bad news for Atletico when they lost their star goalkeeper Jan Oblak to an injury after a collision with Sadiq late in the second half.
"I think we deserved more, we deserved to win," Koke said.
"We played a great game at a very difficult stadium but the ball just refused to go in. It's very frustrating but we have to keep grinding on."
Atletico and Real Sociedad sit sixth and eighth in the LaLiga standings on seven points each, five behind leaders Real Madrid.