Van Persie saw a powerful shot ricochet off the right post in the 65th minute, moments after replacing Wayne Rooney, before hitting the same piece of woodwork four minutes later from the penalty spot.
A scrappy opening half by both teams gave way to a United-dominated second half, with Javier Hernandez firing high over the bar in the 50th with an empty net beckoning.
United midfielder Marouane Fellaini was sent off in the 90th minute after his second yellow card of the match.
United remains top of the group with eight points while Sociedad earned its first point of the campaign.
Elswhere, Real Madrid was held to an entertaining 2-2 draw at Juventus on Tuesday, leaving the Spanish side on the verge of qualification for the Champions League knockout stages.
Juventus dominated in the first period and Arturo Vidal gave the Italian side a deserved halftime lead from the penalty spot after Paul Pogba had been fouled by Raphael Varane.
Cristiano Ronaldo leveled for Madrid with his 21st goal in 16 matches this season and turned provider minutes later as Gareth Bale put Carlo Ancelotti's side in front.
But Madrid's lead lasted just five minutes before Fernando Llorente headed Juventus level.
Madrid dropped its first points but remains top of Group B with 10 points, while Juventus slips to bottom spot after Copenhagen beat Galatasaray 1-0.
And reigning champion Bayern Munich stormed to the knockout stage of the Champions League with a 1-0 victory at Viktoria Plzen on Tuesday.
Substitute Mario Mandzukic rose in the area in the 65th minute, just six minutes after coming off the bench to meet a Philipp Lahm cross and head home the winner.
Bayern made it four wins from four games for a brilliant start to the European season under new coach Pep Guardiola. It tops Group D with 12 points ahead of Manchester City.
It was Bayern's ninth consecutive win in the Champions League over two seasons, equaling Barcelona's record from the 2002-03 season.
Plzen slipped to a fourth straight loss is out of contention.
-AP
