The Dutch striker needed fewer than three minutes to put United in front before Patrice Evra doubled their lead in the 67th minute.
Arsenal, for whom this defeat was their third in five matches, then saw this latest loss made 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 playd well," van Persie told Sky Sports.
"We were a bit sloppy with the chances we did not take, we should have scored more but we got three points.
"It is the first time ever I have played against my former club, but in the end it is about the game and we won it, I am pleased with that."
Victory put United two points in front of previous leaders Chelsea.
But the European champions will return to the top spot if they win at Swansea later on Saturday.
Meanwhile United manager Alex Ferguson was frustrated by a scoreline that did not reflect his side's dominance.
"It wasn't a close game," he said. "I think it was a lack of urgency in the game throughout, far too casual."
But the veteran Scottish boss exempted van Persie from criticism, saying of the opening goal: "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 £24-million 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 van Persie's 10th 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 deep into stoppage time with virtually the last kick of the match, but defeat meant the Gunners were left nine points off the summit just 10 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, but said both United's penalty and the red card shown to Wilshere by referee Mike Dean were harsh decisions.
"It was important not to put ourselves in trouble, which is what we did," said Wenger. "We had a lot of the ball after we conceded the first goal but we did not create a lot.
"Then, when we were 2-0 down and down to 10 men it became difficult, if it was 11 v 11 then we could have come back.
"Yes (Wilshere's sending-off) looked harsh from the stand, the second yellow," he added.
"I think we were a bit unlucky, especially with the penalty decision, we always have a penalty against when we play Manchester United, us so we get used to it," the Frenchman said.
"But overall in a game like this you ought to not put yourself on the back foot when you are one down."
Premier League champions Manchester City, the only unbeaten side in English football's top-flight so far this term, are away to West Ham in Saturday's late kick-off match, having started the weekend level on points with United. – Sapa-AFP.