Victory in the early kick-off match saw United replace Chelsea in first place and that is where they stayed after the European champions were held to a 1-1 draw at Swansea City later on Saturday. In Manchester, Van Persie needed fewer than three minutes to put United in front before Patrice Evra doubled their lead in the 67th minute.
Arsenal's day got worse when midfielder Jack Wilshere, in only his second game back following a long injury lay-off, was sent off for a second bookable offence after he caught Evra on the ankle. "It was a special day, in the end it was about the game and I think we played well," Van Persie told Sky Sports.
United manager Alex Ferguson praised Van Persie's goal and his all-round performance by saying: "It was an ideal start for the Manchester United cause and it was good for Robin. I thought he was fantastic today. His movement, his penetration was fantastic."
Van Persie, playing against Arsenal for the first time since his £24million pre-season move from the Gunners, wasted little time in giving United the lead.
He struck in the third minute when, after Thomas Vermaelen failed to clear a Rafael cross from the right, he shot into the bottom corner of the net from the edge of the penalty area. It was the 29-year-old's tenth goal in 13 United appearances.
United should have doubled their lead shortly before half-time when they were awarded a penalty after Santi Cazorla blocked Ashley Young's cross with his arm. But England striker Wayne Rooney saw his spot-kick go well wide of Arsenal goalkeeper Vito Mannone's right post.
However, Evra did make it 2-0 when he headed in a Rooney cross following a short corner. Cazorla pulled a goal back for Arsenal with virtually the last kick of the match, but defeat meant the Gunners were left nine points off the summit just ten games into the season.
Arsenal manager Arsene Wenger said his side had done themselves few favours in falling 1-0 behind so early in the match. "It was important not to put ourselves in trouble, which is what we did," said Wenger.
"Yes (Wilshere's sending-off) looked harsh from the stand, the second yellow," the Frenchman added. "I think we were a bit unlucky, especially with the penalty decision."
Chelsea held, Spurs beaten
After a goalless first half, Nigeria winger Victor Moses gave Chelsea a 61st-minute lead at the Liberty Stadium with a flick-on following Gary Cahill's header from an Eden Hazard corner. But Pablo Hernandez's equaliser two minutes from time saw the points shared.
"It is a disappointment we weren't able to finish the game off," said Chelsea manager Roberto di Matteo. "It is a shame because we scored the goal and held on until two mintues from the end."
Belgium midfielder Marouane Fellaini scored twice as Everton replaced Tottenham Hotspur in fourth place on goal difference with a 2-2 draw at Fulham. Fulham took a seventh-minute lead when Bryan Ruiz's 25-yard free-kick went in off a combination of post and Everton goalkeeper Tim Howard.
However, Everton equalised through Fellaini in the 55th minute and 17 minutes later he struck a low shot past Australia goalkeeper Mark Schwarzer. Steve Sidwell, though, rescued a point for Fulham with a late strike.
Wigan Athletic won 1-0 away to Tottenham with Ben Watson scoring the only goal when he shot home from close range following a 56th-minute corner.
Elsewhere, Gabriel Agbonlahor scored for the first time in 29 league games as Aston Villa pulled clear of the relegation zone with a 1-0 win against a Sunderland side managed by former coach Martin O'Neill.
Meanwhile strugglers Norwich City won 1-0 at home to Stoke City, with Bradley Johnson's header a minute before half-time the difference.