"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 £24 million (A$37 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, he shot into the bottom corner of the net from the edge of the penalty area.
It was the 29-year-old's 10th goal in 13 United appearances.
Nigeria winger Victor Moses gave Chelsea a 61st minute lead at the Liberty Stadium with a flick-on after Gary Cahill's header from an Eden Hazard corner.
But Pablo Hernandez equalised two minutes from time.
"It is a disappointment we weren't able to finish the game off," said Chelsea manager Roberto di Matteo.
At Upton Park, Manchester City were fortunate to remain the only unbeaten side in English football's top flight this term following a fourth draw in 10 league games.
They nearly fell behind early when West Ham midfielder Kevin Nolan lashed a volley into the net only for his effort to be unluckily ruled out for offside.
And the Londoners suffered more bad luck when another volley, this time from Israel's Yossi Benayoun, hit the post before England striker Andy Carroll added to their frustrations by shooting wide.
Still in the first half, City's Mario Balotelli then wasted a chance to open the scoring by volleying over from eight yards out.
This was the first goalless draw involving City in 42 matches.
Belgium midfielder Marouane Fellaini scored twice as Everton replaced Tottenham Hotspur in fourth place on goal difference with a 2-2 draw at Fulham.
Spurs dropped out of the Champions League positions after Wigan's Ben Watson scored the only goal in a 1-0 win for the visitors at White Hart Lane.
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 boss Martin O'Neill.
Meanwhile, Bradley Johnson headed in the only goal of the game a minute before half-time as strugglers Norwich beat Stoke 1-0 at Carrow Road.
-AAP