The Netherlands international's deflected effort maintained United's advantage at the top of the table, but West Ham's lively second-half performance almost brought what would have been a deserved equaliser.
It took just 33 seconds for the hosts to take the lead as Van Persie's control from Michael Carrick's pass took out Winston Reid and the Dutchman's attempted lob deflected off James Collins before looping over the helpless Jussi Jaaskelainen.
Kevin Nolan was booked for scything down Tom Cleverley before the West Ham captain's near-post flick from Matt Taylor's free-kick rippled the side-netting.
Nolan then failed to apply a finishing touch to Taylor's teasing low cross from the right before Andy Carroll, who had been limping in the early stages, saw a first-time volley fly over Anders Lindegaard's goal.
For all their possession, the Red Devils struggled to create chances for the rest of the first half, although Javier Hernandez's curling shot from outside the box forced a smart save out of Jaaskelainen.
West Ham enjoyed a spell of pressure at the start of the second half and almost replicated the luck United had received when Mohamed Diame's deflected shot soared into the air before dropping off target.
The leaders responded with Cleverley's first-time drive appearing to brush Jaaskelainen's fingertips on its way over before the West Ham keeper made a comfortable save from Van Persie's free-kick.
Jaaskelainen was properly extended on the hour mark, though, as the veteran Finn was forced to dive full length to tip Wayne Rooney's superb curling effort beyond the post.
The Hammers had a good chance with 20 minutes to go when substitute Carlton Cole nodded down Nolan's clever pass, but Taylor mistimed his shot from just inside the penalty area.
Anderson pressed Jaaskelainen into an unconventional save and Hernandez shot tamely at the West Ham keeper before the visitors had their best chance with seven minutes left.
A deep free-kick into the penalty area was flicked on for Cole to control and his shot on the turn from six yards out was parried away by Anders Lindegaard, with Modibo Maiga in an offside position when fluffing the rebound.
Maiga then curled over and Nolan saw a shot blocked before Patrice Evra's under-hit backpass almost presented West Ham with a late chance before his blushes were spared by Jonny Evans as United hung on for another three points.