The hosts played 51 minutes with 10 men after Jonjo Shelvey's sending-off but still managed to take the lead through Steven Gerrard's volley seconds after the interval.
However, Rafael equalised soon after and Van Persie, one of three United players to miss from the spot this term, scored his fourth goal in as many games against the Reds - although three of those had come for former side Arsenal - nine minutes from time.
In the end it was something of a relief the talking points were all generated on the pitch as Liverpool marked their first home match since the publication of the Hillsborough Independent Panel's report which exonerated fans in relation to the 1989 disaster.
Luis Suarez and Patrice Evra shook hands in a show of unity before the game - the Liverpool forward had refused to shake the United defender's hand in the league meeting between the sides at Old Trafford in February, having previously been found guilty of racially abusing Evra in the corresponding fixture at Anfield last season.
Shelvey robbed Ryan Giggs in midfield but as the ball broke free he launched himself at Jonny Evans and the inevitable outcome was a red card.
The visitors would have been pleased to reach the interval with their goal intact but it took just 50 seconds after the restart for that to change.
Second-half substitute Suso, making his Premier League debut three days after his first-team bow in Switzerland, saw his cross half-cleared to Glen Johnson. A tackle from Paul Scholes, on for Nani at half-time, succeeded only in diverting the ball towards Gerrard who volleyed home left-footed in front of the Kop.
It was short-lived, however, as within five minutes United were level when Shinji Kagawa laid the ball off for Rafael to brilliantly curl left-footed over Jose Reina and in off the far post.
Daniel Agger and Johnson collided to allow Antonio Valencia to race 50 yards into the area and although Johnson got back he could only bring down the Ecuador international. Van Persie drilled home - although Jose Reina went the right way and almost made the stop.