The Red Devils bounced back from their humiliating 5-3 defeat to Leicester City last week with a 2-1 victory at Old Trafford as Wayne Rooney and the Dutchman fired Louis van Gaal's men to the three points.
United were forced to dig deep late on when Rooney was sent off for kicking out at Stewart Downing, but Van Gaal's side showed resilience to earn their second win of the season.
"I think this goes even further than three points," Van Persie told the club's official website. "It was absolutely vital to get the three points for everyone's belief and the fans' belief. They were right behind us and were absolutely great.
"I thought we played really well in the first half but in the second it was a completely different game because of the red card. At times it was a bit nerve-wracking but we did ever so well and everyone fought really hard."
The Red Devils are seventh in the Premier League with eight points from their six matches this season.