Rickie Lambert headed the hosts in front after 16 minutes, although van Persie levelled seven minutes later with a fine strike. Ten minutes into the second half, Patrice Evra fell to allow Morgan Schneiderlin to head home.
Saints goalkeeper Kelvin Davis then saved a cheeky penalty attempt by Van Persie, before the Netherlands international provided a late double salvo to secure the three points for the away side.
Saints were getting a lot of joy down the right flank and made the most of it as Jason Puncheon sent a hanging cross to the back post, which Rafael failed to deal with and Lambert headed home after just 16 minutes.
Shinji Kagawa was first to show the visitor's intent as he struck a speculative 20-yard effort that Davis could only push wide. Southampton rode out that storm but could not prevent United drawing level in the 23rd minute.
Antonio Valencia's cross from the right looked like it would be intercepted by Nathaniel Clyne but the Saints right-back slipped and Van Persie took full advantage, controlling with aplomb before firing home.
Daniel Fox's torrid time continued in the second half as Valencia beat him to a pass from Davis, although the Ecuadorian's cross was not converted by Van Persie. Kagawa came close moments before that as United started the second half just as they had ended the first.
That domination counted for nothing, though, as another penalty-box slip resulted in a goal. This time Evra was the unfortunate party as his fall allowed Schneiderlin to nod home a Lambert cross unopposed.
After Van Persie saw an effort saved by Davis, the Saints goalkeeper sent a terrible mis-kick straight to substitute Nani on the left. The ball was worked into the box and Jos Hooiveld conceded a penalty for a challenge on the former Arsenal striker. Van Persie's chip was saved by Davis, much to the delight of the home faithful.
However, Saints could not stop United grabbing a later equaliser as Rio Ferdinand's header came back off the post and Van Persie directed home three minutes from time. Things got worse for the hosts in stoppage time as Van Persie nodded home Nani's cross to ensure the three points would go back to Lancashire.
Source: PA
Source: PA