The Dutchman put United ahead with a low shot midway through the first half and it was his free-kick that set up the second for Nemanja Vidic.
Daniel Sturridge pulled one back for Liverpool in a much improved second-half display but United held out.
Liverpool were oddly timorous at first and had offered nothing when United took the lead after 19 minutes.
A flurry of passes created space for Patrice Evra, who crossed low for Van Persie. He had found a metre of space in front of Daniel Agger, and swept a low finish past Pepe Reina.
Tom Cleverley flashed a volley just wide and Van Persie had a flick cleared off the line by Martin Skrtel.
United doubled their lead 11 minutes into the second half.
Van Persie's free-kick was met at the back post by Patrice Evra, whose header flicked Vidic on its way past Reina.
But within two minutes the visitors were back in it, Sturridge, who had come off the bench to make his Liverpool league debut, following in after David De Gea had saved Steven Gerrard's shot.
Reina made a diving save to deny Shinji Kagawa but the longer the game went in the more dangerous Liverpool appeared.
Sturridge blasted a good chance over with six minutes remaining but United held out.
United at least temporarily moved 10 points clear at the top of the table. Second-placed champions Manchester City were playing at Arsenal in the day's late kick-off.