Van Persie's early effort and a second-half header from Patrice Evra saw United leapfrog Chelsea after the London side conceded a late equaliser in a 1-1 draw at Swansea City.
Victor Moses put Chelsea ahead just past the hour in south Wales but Pablo Hernandez grabbed a point in the 88th minute.
United are a point clear of the European champions with 24 from 10 games while Manchester City missed the chance to move level on points at the top after drawing 0-0 at West Ham United.
Marouane Fellaini scored twice for Everton in a 2-2 draw at Fulham as they rose to fourth place with 17 points, ahead of Tottenham Hotspur on goal difference.
Andre Villas-Boas' Spurs side slipped to a surprise 1-0 home defeat by Wigan Athletic whose scorer Ben Watson was the first Englishman to net a league goal for the club this year.
Norwich City beat Stoke City 1-0 while Aston Villa achieved only their second win of the season, 1-0 at Sunderland.
After a controversial blood and thunder clash at Stamford Bridge last Sunday when United beat Chelsea 3-2 to end the west London team's unbeaten start to the league season they might have expected a similar battle at home to Arsenal.
They need not have worried, however, as the club they once regularly sparred with for silverware produced a listless performance that could have been more harshly punished.
MISSED PENALTY
United, who thrashed Arsenal 8-2 in the corresponding fixture last season, could even afford a missed penalty by Wayne Rooney on the stroke of half-time.
Santi Cazorla's well-taken consolation goal in stoppage time flattered the visitors who had midfielder Jack Wilshere sent off for a second booking after the break.
All the more galling for Arsenal was the sight of Van Persie scoring against them after eight years of banging in goals for the North London club who are losing touch in the title race.
The Dutchman, who joined United from Arsenal in August having scored 37 league goals last season, needed less than three minutes to score after a mistake by Thomas Vermaelen.
Van Persie rarely needs help finding the back of the net but central defender Vermaelen obliged when he made a hash of trying to clear a cross from Rafael and the ball fell to his former team-mate who struck a first-time shot past Vito Mannone.
"A special day, in the end it was quite a game," Van Persie, who did not celebrate his eighth league goal since joining United, told Sky Sports.
"It speaks for itself [why I did not celebrate]. I played eight years there and had a fantastic time. I wanted to respect the fans, the manager and the whole club."
Arsenal struggled to make any real impact and only Wilshere, making his second start since returning from a long injury lay-off, showed any bite in the opening period - earning a yellow card for
