Van Persie converted a late penalty kick to clinch a 2-1 victory at Anfield after both teams had paid tribute to the victims of the 1989 Hillsborough disaster before the game.
United was a man up from the 39th minute when Jonjo Shelvey was sent off, but Steven Gerrard managed to give Liverpool the lead with a volley at the start of the second half — only for Rafael da Silva to equalize.
And Van Persie's winner prolonged Liverpool's worst start to a season in more than a century, while lifting United to second.
City is three points further back after Joleon Lescott headed the champions ahead in the first half, but then failed to clear a late corner, allowing Laurent Koscielny to recover a 1-1 draw for Arsenal.
Also Sunday, Jermain Defoe's winner gave Tottenham a 2-1 win against Queens Park Rangers and Demba Ba gave Newcastle a 1-0 victory over Norwich.
The day's action began with an emotionally charged meeting of English football's fiercest rivals as Liverpool played at Anfield for the first time since the release of a report into the Hillsborough tragedy that exposed the police coverup and exonerated Reds fans.
In a show of unity before kickoff, Liverpool and United players wore tracksuit tops featuring the number 96 — the amount of Reds fans crushed to death 23 years ago.
Luis Suarez even shook hands with Patrick Evra, with no repeat of the snub that followed the Liverpool striker being banned for racially abusing the United defender in the Anfield fixture last October.
But by the end of the game there had been chants from a small number of Liverpool fans about the 1958 Munich disaster which killed eight United players. And, despite pre-match appeals from manager Alex Ferguson, there were taunts from United fans directed at the home support.
The usual hostilities had started to rage after Shelvey was sent off for launching himself at Jonny Evans, who Liverpool manager Brendan Rodgers felt should also have been dismissed.
"I think both players' feet are off the floor," he said.
There was also a heated exchange of words between Shelvey and Ferguson, who later said: "He was trying to blame me."
There was an apology — of sorts — from Shelvey later.
"I have also apologised to sir Alex, just where I come from people don't grass people up to get someone sent off," Shelvey wrote on Twitter.
Gerrard put Liverpool in front within 50 seconds of the second half, after smoothly bringing the ball down with his chest before volleying into the net.
But within five minutes, Shinji Kagawa fed Rafael to curl the ball beyond goalkeeper Pepe Reina.
And with 10 minutes to go, United winger Antonio Valencia was brought down by Glen Johnson in the penalty area, and Van Persie netted despite Reina getting a touch to the spot kick.
"With the occasion and the emotions involved, we just wanted to come here, get the job done and get out with the three points," Ferguson said. "We have done that so we are well pleased. You can get really involved and sometimes the emotion of the occasion can affect you."
At City, David Silva's corner was headed past goalkeeper Vito Mannone, playing in place of the injured Wojciech Szczesny, five minutes before halftime by Lescott.
Santi Cazorla had only just given City goalkeeper Joe Hart his first meaningful save to make when he took the corner that Lescott diverted to Koscielny, who blasted in the leveler in the 82nd.
"I don't know if we lost our concentration but it's strange," City manager Roberto Mancini said. "We had the game under control."
At White Hart Lane, former Tottenham striker Bobby Zamora gave QPR a halftime lead, but Alejandro Faurlin's own goal on the hour handed the initiative to the hosts.
Inside a minute, Defoe tapped in eighth-place Tottenham's winner.
Tottenham is level on points with Newcastle, which relied on Ba's third goal in two games to ensure Norwich is still yet to win this season.Suggest a correction