Van Persie rose unchallenged to guide home Wayne Rooney's 27th-minute corner to spark scenes of jubilation at Old Trafford and silence the taunts of the visiting fans.
United have now closed to within five points of Arsenal, who suffered their first league defeat since the opening day of the season.
Indeed, six points now separate the first eight in the table, with Liverpool and Southampton now Arsenal's closest pursuers.
United deserved their triumph too.
Rooney ran himself to a standstill and Phil Jones was immense, initially in midfield, then central defence following the departure of Nemanja Vidic with a head injury.
The hosts owed a debt of thanks to Chris Smalling too.
Smalling might have wasted an outstanding chance to seal victory when he failed to get a proper connection to Van Persie's deep free-kick and headed disappointingly wide.
But, just as Olivier Giroud was lining up a shot from an excellent position inside the United box, it was Smalling who stepped in to make the decisive tackle and preserve United's unbeaten home record against the Gunners that stretches back to 2006.
There was no real surprise Van Persie should return to the United starting line-up, still less the Dutchman celebrated the opener with intent.
The trend has been for players to merely acknowledge what they had done, rather than glory in the pain they had caused to their old club.
Van Persie refrained himself when he scored in the corresponding fixture last term.
However, the taunts aimed at him by the visiting contingent throughout the opening stages at Old Trafford evidently changed his stance.
For Van Persie raced away in glee as his header flew over Kieran Gibbs before he was submerged by team-mates, then adopting a clenched fist salute as he re-appeared.
The blame for Van Persie's goal seemed to be pinned on Aaron Ramsey, who moved away from his station as Rooney delivered the corner.
There had to be some culpability as it was surely not part of Arsene Wenger's game-plan to afford such a predatory striker so much room.
It was not the only issue for Wenger to address at half-time.
Since that shock opening-day defeat to Aston Villa, that first period was as bad as Arsenal have been.
Missing Per Mertesacker due to a virus that was also affecting Mikel Arteta, the Gunners lacked the zip that has taken them clear at the Premier League summit.
Only Mathieu Flamini escaped the malaise, and he was booked for a late tackle on Jones.
The young England star was one of United's stand-out performers, driving out of midfield in a manner so obviously lacking for the past three seasons.
He was also at the centre of the first of two worrying head clashes.
Cautioned when he clattered into Wojciech Szczesny, TV replays showed the contact to be accidental.
For a few seconds, the Arsenal keeper did not move.
After treatment, Szczesny continued, as did Jones, who had moved away rubbing his head.
Nevertheless, given the recent furore over Hugo Lloris, the decision raised an eyebrow.
In the final seconds of the opening period, Vidic went down in a heap after his face was crunched by David de Gea's knee.
There was no immediate substitution as United reached the interval with 10 men.
The tough-tackling Serbian did not return for the second period though as Moyes introduced Tom Cleverley and moved Jones back into central defence.
It all rather obscured a disappointing lack of goalmouth action.
Szczesny had earlier saved a long-range effort from Rooney but Arsenal were devoid of opportunity as Olivier Giroud failed to find any space and Mesut Ozil struggled to make an impact.
The visitors improved after the break, with Ozil slicing an angled effort into the side-netting.
Wenger still felt compelled to introduce Jack Wilshere after an hour, a move that came just after his side had survived Rooney nipping past Bacary Sagna and belting a low shot inches wide.
The visitors pushed forward in increasing desperation. But United defended valiantly.
Smalling was involved in key moments at both ends before Sagna's low cross bounced inches away from Nicklas Bendtner.