Louis van Gaal's side picked up yet another three points thanks to their Netherlands international striker as the Saints lost for the third match in a row
The Dutchman took advantage of some poor defending to open the scoring in the 12th minute although the hosts were back on level terms on the half-hour mark as Graziano Pelle lashed home from close range.
However, Van Persie was on hand to give his side all three points, directing Wayne Rooney's free kick home as Louis van Gaal's side leapfrogged the Saints in the table.
|VIEW FROM ST MARY'S
|By Liam Twomey
Louis van Gaal's Manchester United are now on a five-game win streak, but Ronald Koeman will be wondering how.
Southampton dominated this Monday night clash pretty much from start to finish, only for a third Dutchman to make the decisive contribution.
Robin van Persie looks to be slowly putting his early-season struggles behind him, scoring clinically in each half from United's only two shots on target and curling another just beyond a diving Fraser Forster's post.
On this form, United will need Van Persie's ruthlessness to bail them out for some time yet, even as results suggest momentum is building. Southampton, meanwhile, are still waiting to crown their remarkable rise this season with a win over a member of the Premier League's established elite.
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 to fizzle 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.