A poignant day at Anfield started with both sets of fans behaving impeccably during a tribute to the 96 supporters killed in the Hillsborough tragedy but Liverpool's gloomy season continued when Van Persie rammed home a late spot-kick after Glen Johnson had brought down Antonio Valencia.
Liverpool, reduced to 10 men in the first half when Jonjo Shelvey was red-carded, had led through Steven Gerrard's goal just after half-time but Rafael equalised for United who are second in the table with 12 points, one behind Chelsea.
There were grumbles of discontent at the Etihad Stadium as City drew for third time in their opening five games, Laurent Koscielny smashing a late equaliser for Arsenal after Joleon Lescott had headed the hosts into a half-time lead.
Both sides are unbeaten with nine points, although Mancini was clearly frustrated.
"This year we have one problem, we don't close the game and don't score more goals," he said "We had three or four chances in the second half and didn't score."
The most blatant fell to Sergio Aguero who missed an open goal after Vincent Kompany's bicycle kick had been saved.
"I am frustrated about the result. We need to work harder."
Tottenham Hotspur have eight points after Jermain Defoe sealed a 2-1 victory over Queens Park Rangers, Spurs's first home win under new manager Andre Villas-Boas.
Demba Ba's early goal earned Newcastle United a 1-0 win over their former manager Chris Hughton's Norwich City.
The first match at Anfield since the Hillsborough report revealed a police cover-up and cleared Liverpool's fans of blame for 96 of them being crushed to death at a 1989 FA Cup semi-final sparked into life after 39 minutes when Shelvey lunged in on Jonny Evans.
But Liverpool rallied strongly and deservedly went ahead through Gerrard's precise volley.
A lacklustre United responded with Rafael curling a shot beyond Pepe Reina and Liverpool were left to curse their luck when Van Persie kept his cool to beat the despairing dive of Reina for his fifth league goal of the season.
WORST START
With just two points from five games under new manager Brendan Rodgers, Liverpool have suffered their worst start in the league for 100 years, although United manager Sir Alex Ferguson admitted his side had been fortunate to win at Anfield for the first time since 2007.
"I thought we were poor to be honest with you," Ferguson told United's website. "I think the last four years we've allowed the crowd to get to us a little bit - they give fantastic support to their team and they really dominated the first half. Second half they got a great start.
"In the last four years here we haven't played well," he added. "Today at least we've got a result."
One of the fiercest rivalries in world football was briefly put to one side before kick-off with an emotional tribute to the victims of the Hillsborough disaster in the light of this month's independent repo
