The newly promoted side succumbed to a late fightback from the visitors in heartbreaking fashion in a game that, on the whole, they had dominated, having twice been ahead
Rickie Lambert gave the hosts an early lead before Van Persie equalised on the half-hour mark for his second goal in as many starts for his new side.
Morgan Schneiderlin headed his side back in front soon after the break before Van Persie first missed a penalty and then completed his hat-trick by first reacting first to Rio Ferdinand’s header coming back off the post and then heading come from a corner two minutes into added time.
Anders Lindegaard was given the nod by Sir Alex Ferguson to take his places in between the posts, with Ferdinand and Nemanja Vidic holding the fort together in the centre of defence for United, with Nigel Adkins opting to hand 17-year-old midfielder James Ward-Prowse only the second start of his career.
After a nervy opening 15 minutes where United appeared comfortably on top, the hosts pounced.
Schneiderlin got the better of Shinji Kagawa in midfield before finding James Puncheon on the corner of the penalty area. A quick turn later and the former Blackpool attacker sent a hanging cross towards Lambert to head back across Lindegaard to give Southampton a surprising lead.
United responded well, with Kagawa trying to make amends almost immediately with a strong, low drive from 25 yards, which Davis struggled to contain, before the Japan international started the move that resulted in a the away side's equaliser.
Twisting to avoid Steven Davis in midfield, Kagawa found Danny Welbeck trotting towards goal. With his path blocked by red shirts, the striker – in for the injured Wayne Rooney – played in Antonio Valencia on the right wing to send a lofted cross towards Van Persie at the back post. A slip by Southampton right-back Nathaniel Clyne then gave the new signing time to set himself and stroke the ball past the helped Davis.
Whereas United were in rampant mood during weekend's 3-2 victory over Fulham after going behind early on, Southampton were not only able to keep their opposition contained but create many threatening attacks of their own – with Lambert, Puncheon and Adam Lallana pulling the strings.
United started to kick into gear as the first half drew to a close, with first Van Persie with a bending free kick from the edge of the area before Patrice Evra's header from another fine Valencia delivery but they could not find the decisive finish before the referee’s whistle.
The away side continued their new-found impetus into the second half with Kagawa and Van Persie combining well in the opening minutes for the former to flash a shot wide of goal.
However, Southampton again took full advantage of the profligacy of United by going back in front with another well-worked move. This time Lambert turned provider and Schneiderlin, skipping away from Evra who lost his footing inside the area, headed home past Lindegaard.
United were given a lifeline in just minutes later, however, when Valencia pounced on Davis' poor goal kick to find Kagawa inside the area – only for Jos Hooiveld to lunge in recklessly from behind to concede a penalty.
Van Persie duly stepped up and attempted to chip Davis, who had enough time to dive to the ground and flick the ball away for a corner.
However, Van Persie was to have the last laugh when he found himself in the right place at the right time to capitalise on Ferdinand's header pinging back off the post to draw level late on.
As the game headed into extra-time, Southampton were hanging on – in drastic contrast to the rest of the game – and eventually succumbed to Van Persie once more as he latched onto Nani's late corner to complete his hat-trick and tie up the win late on in, fittingly, Sir Alex's 1000th league game in charge.