The Dutch striker netted within 33 seconds to put the Red Devils in front after his first touch took him away from Winston Reid while his shot deflected off James Collins and ended in the back of the net.
United failed to kill the game off despite creating several clear cut chances upfront. The Red Devils are just a point above city rivals Manchester City in the league standings.
Van Persie, who joined United in the summer for £24m, scored his ninth league goal this season. The 29-year-old also pipped his former Arsenal team-mate Theo Walcott who had scored in 53 seconds against Everton on Wednesday.