Chicharito also missed a golden opportunity as David Moyes' side let the chance to take control of its Champions League group slip in Spain.
Robin van Persie missed a penalty and Marourane Fellaini was sent off as United saw its four-game winless run away from home in the Champions League continue.
The point was Real Sociedad's first of this season's competition and keeps its slim hopes of qualifying for the knockout stages alive given Bayer Leverkusen and Shakhtar Donetsk also drew. United remains on top of Group A with eight points, but will feel it should have taken all three with Javier Hernandez also guilty of missing an open goal.
The hosts made four changes from their 5-0 victory over Osasuna on Saturday as Markel Bergara, Alberto de la Bella, Mikel Gonzalez and Imanol Agirretxe all started having been on the bench over the weekend.
David Moyes opted to field Hernandez up front with Wayne Rooney as opposed to van Persie, naming the Dutchman among the substitutes. Hernandez was one of six changes for United from its victory over Fulham in the Premier League.
The visitors started confidently and pressed Sociedad high up the pitch, creating the first chance of the match in the 10th minute. Shinji Kagawa released Patrice Evra down the left side, the Frenchman found Rooney inside the penalty area, and he flicked the ball back to the Japan international to fire a deflected shot wide.
It took the best part of half an hour for Sociedad to create an opening that came after David Zurutuza broke down the right. The midfielder pulled his cross back to Agirretxe on the penalty spot, but he skewed his effort well wide of David de Gea’s goal.
United should have taken the lead early in the second half, but Hernandez was guilty of missing an open goal in the 50th minute. Kagawa shifted the ball past Carlos Martinez and delivered a low cross into the unmarked Mexican, but he ballooned his effort over the bar from 6 yards.
The visitors’ frustration continued just after the hour as van Persie – introduced in place of Rooney – struck the post with his first touch after an excellent cross from fellow substitute Ashley Young.
The Netherlands international striker was given another opportunity to score four minutes later after Bergara was judged to have pulled back Young inside the penalty area. Van Persie’s spot kick was parried away by goalkeeper Claudio Bravo after it struck the post.
Fellaini spoiled United's night with his late dismissal after a second yellow card following a challenge on Zurutuza.