After a dismal first half, Javier Hernandez inexplicably failed to tap home Shinji Kagawa's low cross from barely six yards before substitutes Van Persie and Ashley Young took centre stage.
Within minutes of their joint arrival, Young provided the cross which Van Persie belted against a post.
More contentious was the spot-kick awarded to the visitors when Young went down under Markel Bergara's challenge.
Although referee Nicola Rizzoli was certain, others were less so.
Van Persie was not worrying about that as he strode up to take the kick. He soon had cause to regret his own failings, though, as his effort also bounced back off the woodwork before clipping goalkeeper Claudio Bravo on the way out and rolling to safety.
It was the first time Van Persie had missed from the spot since his failure at Southampton in the first weeks after his arrival from Arsenal. He atoned for that miss by scoring a match-winning hat-trick.
There were no such heroics here and with Fellaini red-carded for the first time in his United career for a second yellow card, the visitors were forced to see out the contest with 10 men.
However, with the other game in the group also ending goalless, no damage was done to United's chances of reaching the last 16, which they will reach with a win from either of their two remaining fixtures.
Though David Moyes insisted in his pre-match briefing there was nothing different in his side's approach to their European and domestic engagements, progress at this elite level has been achieved far more comfortably than it has in the Premier League.
With seven points accumulated already, he could even afford to leave Van Persie on the bench, allowing Hernandez an unexpected start, with two more men on the fringes, Kagawa and Fellaini, joining the Mexican.
Not that any of them could raise the standard in what was a mediocre opening period.
There was plenty of industry, with Wayne Rooney darting back well inside his own half to rob Carlos Vela after his misplaced pass gifted possession to the former Arsenal player.
Chris Smalling also put a shift in at right-back whilst, on his 950th appearance, Ryan Giggs tried to cajole a bit more from the visitors.
It was not great stuff, though, and UEFA's own statistic of three shots in the first half-hour had observers scratching their heads to remember them.
Kagawa was responsible for one, at the end of the best move of the match.
Patrice Evra delivered a superb low cross into the box, which Rooney flicked back towards the Japan star.
Kagawa's shot was tame, though, and would not have troubled the Socieded goal even without the deflection that took it wide.
Fellaini brought a bit of life to proceedings when he caught Markel Bergara with a tackle that upset his opponent and earned the Belgian a yellow card.
The hosts were even less inspiring, a major surprise given the quality within their ranks and the fact they had amassed zero points thus far.
Alberto de la Bella eventually curled a shot wide as the drab opening period limped towards its conclusion.
In Hernandez's inexplicable miss alone, the second half contained more to discuss than had gone before the break.
With three goals in his last two games, prior to being left on the bench at Fulham, it is not as if he was out of form.
There certainly appeared no plausible excuse for his woeful miss, skying high over the net from only six yards after Kagawa's cross had rolled perfectly into his path.
Hernandez was replaced not long afterwards as part of a double substitution that also saw the exit of Rooney as Van Persie and Young were introduced. Moments later, the latter was crossing for the former, from much the same spot as Kagawa had delivered for Hernandez.
Van Persie's angle was acute. The power behind his shot was impressive, though, as the ball shot back off the near post. The Dutchman seized on the rebound but from 20 yards could only curl wide.
Finally, though, the game had sprung to life. And when Young went to ground - too easily for some even though Bergara was booked - Van Persie seemed sure to score from the spot.
He failed. And as Socieded countered Xabi Prieto's goalbound effort was deflected over.
The hosts got no closer, even after Fellaini had departed in the final minute.
Source: PA
Source: PA