Van Persie was given the chance to seal a win against the Basque side after Ashley Young fell easily in the area under the challenge of Markel Bergara, but he smashed his kick against the post.
Marouane Fellaini was sent off in the closing stages, after picking up a second yellow card for a brainless challenge on David Zurutuza, forcing David Moyes' men to cling on to a point when they might have won the game.
United showed surprisingly little ambition or purpose in the first half, only carving out two chances – both shots by Shinji Kagawa - but they were often too easily pressed back into their own half by a Sociedad side that grew in confidence as the game progressed.
Indeed, the hosts looked the most likely to take the lead in the closing stages of the first half. Nemanja Vidic was forced into a late saving tackle as Carlos Vela latched onto an excellent Carlos Martinez through-ball, and on the stroke of half-time Alberto de la Bella almost scored after being allowed to easily cut inside from the left, his right-footed shot going just wide of an unsighted David De Gea's post.
The English champions improved after the break, and should have scored not long after half-time, when Kagawa carved out a chance for Javier Hernandez on the edge of the six-yard box, but the Mexican blazed over.
Van Persie was brought on as a substitute and nearly had an instant impact, hitting the post with his very first touch. The Dutchman then missed from 12 yards, leaving United frustrated.
United remain unbeaten in the Champions League, top of Group A after the other game in the group between Shakhtar Donetsk and Bayer Leverkeusen also ended 0-0.
PRE-MATCH - There are rare starts for Shinji Kagawa - which will either provide evidence for those who want him in the United team or expose that his reputation has risen beyond itself in absentia - and Javier Hernandez, who'll probably poach a goal early on and celebrate like a 2003 vintage Inzaghi.
38 - GREAT CHANCE FOR UNITED - Nearly for United - Kagawa wriggles his way into a bit of space on the left, he drills a low cross to the near post and Inigo Martinez just gets there to cut the ball out ahead of Hernandez.
45 - CLOSE FOR SOCIEDAD - De La Bella nearly gives Sociedad the lead by cutting in from the left and shooting with his right foot. De Gea was unsighted and it just goes wide of the post.
50 - WHAT A MISS! - Kagawa does well to shift some space in the area and finds Hernandez on the edge of the six yard box, but the Mexican scoops the ball over the bar. It might've taken a slight bobble, but that's still a bad miss.
69 - PENALTY MISSED! Van Persie slams the kick into the post and it flies out. A little justice, perhaps.
90 - FELLAINI IS DISMISSED! Well, that was stupid. Fellaini gets a second yellow card for a brainless trip on Zurutuza.
Manchester United had 55% possession and completed 81% of their passes.
Real Sociedad had a remarkable 29 throw-ins throughout the game.
Sociedad didn't have a single shot on target.
TALKING POINT: United's approach - While Manchester United were perhaps a little unlucky not to win after Robin van Persie's penalty miss, the way they approached the game in the first half was curious. Facing a clearly inferior Real Sociedad side, they seemed to offer little going forwards, and indeed were almost caught out as the hosts attacked with purpose as they gained confidence. Had United asserted themselves early on, they might not have needed to hang on for a point in the closing stages.
MAN OF THE MATCH
Shinji Kagawa (Manchester United): Comfortably United's biggest threat, Kagawa produced a few good chances when playing from the left, and looked even more threatening when moved inside in the second-half. Food for thought for David Moyes.
REAL SOCIEDAD: Bravo 6; C. Martínez 8, Mikel 6, Iñigo 7, De la Bella 7; M. Bergara 6, Pardo 6, Zurutuza 6; Vela 7, Agirretxe 7, Griezmann 6. SUBS: X.Prieto 6, Chory Castro 5, Seferovic 5.
MANCHESTER UNITED: De Gea 7; Smalling 6, Ferdinand 7, Vidic 6, Evra 6; Valencia 5, Fellaini 4, Giggs 6, Kagawa 8; Rooney 6, Hernandez 5. SUBS: Jones 5, Young 5, Van Persie 6.
THE MATCH IN A TWEET
- Sports & Recreation
- Manchester United
- Real Sociedad
- Shinji Kagawa
- Robin van Persie
- David Moyes
- Marouane Fellaini
- David Zurutuza
- David De Gea