The prolific Dutchman sealed the victory when he fired his spot kick past the despairing dive of Pepe Reina after Glen Johnson had brought down Antonio Valencia.
Despite being well below their best United moved into second spot with 12 points, one point behind leaders Chelsea.
The first match at Anfield since the Hillsborough report cleared Liverpool’s fans of blame for the deaths of 96 of their number at the 1989 FA Cup semi-final, sparked into life after 39 minutes when Jonjo Shelvey was red-carded for a dangerous lunge.
However, the home side deservedly went ahead straight after halftime through Steven Gerrard’s volley.
United hit back after 51 minutes through Rafael’s curler before Van Persie’s late strike meant that after five games Liverpool are still looking for their first league win of the season – their worst start to a Premier League campaign.
“The last three or four years we have come and played better but not got anything, that is the way it goes sometimes,” United skipper Ryan Giggs told Sky Sports.
“Liverpool wanted to put on a performance because of the occasion and they did that. They played well, but we can play better.”