The widely documented frosty relations between Southampton manager Ronald Koeman and his compatriot and counterpart at United Louis van Gaal dominated the pre-match build-up, but it was the Dutchman on the field who stole the show at St Mary's Stadium.
Van Persie's first goal was a 12th-minute gift after Southampton captain Jose Fonte's careless backpass and he benefitted from slack Southampton play with 20 minutes to play, prodding home after Fraser Forster and Victor Wanyama were caught napping under Wayne Rooney's lofted free-kick.
In between Van Persie's goals, Southampton were largely the most threatening team and their attacking intent was rewarded when Graziano Pelle netted a first-half equaliser - his 10th goal in all competitions this term.
As Shane Long and Pelle both passed up chances to find a winner for the hosts, Van Persie condemned them to a third-straight loss and secure bragging rights for Van Gaal.
Both teams probed for an opening without success during cautious opening exchanges before Fonte endured a moment to forget on his 200th Southampton appearance.
Under no more than gentle pressure from returning United captain Rooney, the Portugal defender under-hit a backpass and Van Persie needed no invitation to collect the loose ball and slot calmly past Forster.
United's defensive injury woes struck again in the 16th minute - Chris Smalling hobbling off with an apparent groin strain and replaced by Jonny Evans on his comeback from an ankle complaint.
Sadio Mane looked lively on Southampton's left after a recall to Koeman's XI, pouncing on Paddy McNair's heavy touch and storming into the United area, only for the chance fizzled out before Pelle tested David de Gea from Dusan Tadic's cutback.
Southampton were posing increasing questions and they proved too taxing for Marouane Fellaini in the 31st minute as he carelessly gave away possession to Steven Davis.
The Northern Ireland international drove toward the United area, with the ball eventually finding the feet of Pelle after defensive dithering and the Italian slammed home.
Long blazed over from close range as United displayed further defensive uncertainty, prompting Van Gaal to reverse a change to his starting XI and haul off McNair in favour of Ander Herrera six minute before the interval.
United were on the back foot once more when the action resumed and Southampton should have taken a 52nd-minute lead.
Tadic's teasing cross again exposed vulnerabilities on the left-hand side of Van Gaal's defence and Long should have done better than planting a header into De Gea's midriff.
Southampton continued to push on so it was therefore against the run of play when Van Persie directed Rooney's delivery into the net with the outside of his boot for the winner.
De Gea added to his catalogue of magnificent saves over recent outings in the 77th minute - the Spain international plunging to his left when Pelle fizzed a strike goalwards on the turn.
United leapfrog Southampton and go third ahead of West Ham, lying six and eight points behind champions Manchester City and leaders Chelsea respectively.
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