Van Persie beat Liverpool goalkeeper Pepe Reina in the 81st minute after Antonio Valencia was fouled by Glen Johnson to secure United's first win at Anfield since December 2007.
"It has been four or five years since we got a win here. It was an ugly win, probably our worst performance in that time, but we will take that," United manager Alex Ferguson said. "We didn't get going as an attacking force. We were on the back foot for most of the first half and sometimes it creates its own problems."
Liverpool played with 10 men for more than 50 minutes after Jonjo Shelvey's sending-off before half time but still managed to take the lead through Steven Gerrard's volley within a minute of the second half starting.
Defender Rafael da Silva equalized five minutes later, curling the ball past Reina, and Van Persie's winner ensured second-place United prolonged winless Liverpool's worst start to a season in more than 100 years.
"I thought the players were heroic in terms of performance and the spirit," Liverpool manager Brendan Rodgers said. "They were brilliant and didn't get what they deserved."
The build-up to the match, though, had been about the fallout from the report into the 1989 Hillsborough tragedy that absolved Liverpool supporters of any responsibility and exposed a police coverup.
Old rivalries between Liverpool and United were put aside before the first match at Anfield since the report was published, with both sets of players emerging on the pitch wearing tracksuit tops featuring the number 96, the number of Reds fans crushed to death 23 years ago.
Liverpool striker Luis Suarez and United defender Patrick Evra ensured a reconciliatory start as they joined the other players in a pre-match handshake. Last season Suarez refused to shake Evra's hand when the teams played at Old Trafford after the Uruguay international was banned for racially abusing his United rival.
For the first minute of the match three sides of Anfield displayed mosaics featuring "The Truth," ''Justice" and "96."
The tributes seemed to inspire Liverpool's players on the pitch, with domination from the hosts.
Once Ryan Giggs had flashed an early shot wide the momentum was all with Liverpool as Suarez's cross-shot was stopped by Anders Lindegaard and headed away by Jonny Evans.
Gerrard drove a low effort from Shelvey's corner into the side-netting and Suarez looped a right-footed shot wide before the usual hostilities in the north-west derby ignited.
Shelvey seized possession but as the ball broke free he launched himself at Evans and was shown a red card. Rodgers, however, said both players should have been given the same punishment for that duel.
"Jonjo Shelvey, if he gets sent off then Jonny Evans has to go as well," Rodgers said. "I think both players' feet are off the floor."
Ferguson was involved in a heated exchange of words as Shelvey left the pitch.
"He was trying to blame me, I suppose," Ferguson said. "Once he looks at it he can apologize if he likes."
Shelvey, though, mainly only apologized to the fans.
"No way was I pulling out of that tackle in a game of that importance. Im sorry," Shelvey wrote on Twitter. "I have also apologised to sir Alex, just where I come from people don't grass people up to get someone sent off."
Even with a numerical disadvantage, Liverpool continued to pressure United and Suarez drew Lindegaard into a low save from his free kick.
And at the start of the second half, Gerrard received a diverted ball in the area and volleyed home left-footed in front of the Kop.
The lead only lasted five minutes, however, as Shinji Kagawa laid the ball off for Rafael to curl left-footed over Reina and in off the far post.
But, even with a man down, Liverpool pushed for a winner with Suso having a shot tipped over by Lindegaard and Suarez continuing to trouble the United defence.
United began to take more control, though, and the winner came from Liverpool giving away possession on the halfway line.
Daniel Agger and Johnson collided to allow Valencia to race 50 yards into the area and, although Johnson got back, he brought down the Ecuador winger. Van Persie drilled home despite Reina diving the right way and getting a hand to the ball.Suggest a correction