The Saints had lead twice through Rickie Lambert and Morgan Schneiderlin, but van Persie, who also missed a penalty, struck goals in the 24th, 87th and 92nd minutes to give United manager Alex Ferguson the perfect present as he marked his 1000th league match in charge of the Old Trafford club.
Elsewhere Newcastle were held to a 1-1 draw at home by Aston Villa. Hatem Ben Arfa's 59th-minute goal cancelled out Ciaran Clark's opener.