The Senegal international cut the ball back from the byline for the unmarked striker to sweep the ball past David de Gea.
Wayne Rooney tried to get United back on level terms with a right-footed shot from just outside the area, but Mario Felgueiras saved easily in the end.
United did restore parity just before the half-hour mark, when van Persie turned in Rooney's free kick with his shoulder.
The decisive moment came in the 49th minute, when the Dutchman doubled his tally with a superb finish from Rooney's pass.
De Gea made a terrific save from Rafael Bastos late on to ensure that United took all three points.