The Dutch international had only been introduced by United boss David Moyes as a substitute, but saw his spot-kick come back off Claudio Bravo's left-hand post 20 minutes from time.
Javier Hernandez and Van Persie had also passed up clear opportunities to put the visitors in front before the penalty, but they had to see the game out with 10 men after Marouane Felliani was shown a second yellow card in stoppage time.
Another 0-0 draw between Shakhtar Donetsk and Bayer Leverkusen means United continue to lead the Germans by a point with the two sides set to meet in Leverkusen in three weeks time.
Moyes was without the injured duo of Jonny Evans and Rafael da Silva, whilst teenage sensation Adnan Januzaj was also let out of the squad and Van Persie dropped to the bench with Sunday's meeting against his old club Arsenal in mind.
However, United still started the better of a largely lifeless first-half. Shinji Kagawa had the best opening with an effort that was deflected wide from a lovely Wayne Rooney backheel on 10 minutes.
Sociedad improved as the half went on and Alberto de la Bella came within inches of opening the scoring with the last kick of the half as his right-footed effort curled just wide.
Hernandez then somehow conspired not to give the visitors the lead five minutes after the restart when he blasted Kagawa's low cross over the bar from five yards.
Kagawa then had a penalty appeal waved away moments later as he clashed with Mikel Gonzalez inside the area. With United still in search of a goal Moyes sent on Van Persie and Ashley Young for Rooney and Hernandez just after the hour mark.
And the Dutchman nearly made an instant impression as he firstly side-footed Young's low cross off the post then flashed another effort just wide. Referee Nicola Rizzoli then incensed the home fans when he pointed to the spot for a very soft pull on Young by Markel Bergara.
However, the hosts were handed a reprieve when Van Persie's fiercely struck spot-kick came back off the post. And Sociedad nearly made the most of it when Xabi Prieto's drive was turned goalwards by Carlos Vela but flew inches past the post.
Antonio Valencia's effort was cleared off his own goal line by Gonzalez as the game continued to open up in the final stages. But even after Felliani had been dismissed Sociedad failed to create another clear opening and had to settle for their first point of the group.