The Dutchman, who left the Emirates in the summer, fired home early on in his first meeting against his old employers as United proved too strong for Arsenal.
Van Persie felt United should have won more convincingly, but he was happy in the end to take all three points.
"It was a special day but in the end it was about the game," Van Persie told Sky Sports 1HD. "I think we played well, we were a bit sloppy with chances and should have scored a few more but in the end I'm happy with the three points."
Van Persie also revealed he chose not to celebrate his goal against his old club out of respect for his former employers.
"I think it speaks for itself," added Van Persie.
"I played for eight years and had a fantastic time there. I respect the fans, the players, the manager and the whole club.
"It's the first time ever that I've played against my former club because I only have three. It was a special day but in the end it's about the game and we won it."
Van Persie also played down any problems with any of the chants aimed his way by some Arsenal fans.
"I honestly didn't hear them. They are great fans and I do respect them a lot," concluded Van Persie.
Van Persie's strike partner, Wayne Rooney, conceded his side need to be more clinical and finish games off after dominating for long periods of the game.
"I think we had plenty of chances but we should be taking care of games," said Rooney.
"We had a lot of chances late doors, especially when they were down to 10 men to kill the game off and we couldn't.
"That could cost us in the future so we need to make sure we're more clinical and finish the games off."
Rooney was guilty of missing a golden opportunity when he fluffed his penalty on the stroke of half-time after Santi Cazorla was adjudged to have handled inside the box.
"These things happen," Rooney said of his penalty. "Obviously I'm disappointed with the penalty but I'll move on.
"We won the game, that's the most important thing, the penalty is something that is gone for me now."
The win lifts United above Chelsea into first place in the Premier League table and Rooney was happy to go to the summit for a few hours at least.
"We knew before the game if we won we'd go top of the league and it's important for us," continued Rooney.
"After the first day of the season and dropping three points we've done well. We've grinded results out and we've played some good football."