For only the second time this season Sean Dyche's side finished without a point after scoring the first goal of the game.
Burnley - who were beaten 4-2 by Leicester City after Chris Wood had opened the scoring at the King Power Stadium in September - had been cruising on the south coast.
The New Zealander struck first against the Saints when beating Fraser Forster from the spot in the 12th minute.
And the visitors extended their lead just before the half-hour mark when Matej Vydra finished from outside the box from Wood's flick on.
However, Southampton responded well and were back on level terms before the break.
Stuart Armstrong fired past Nick Pope from a Danny Ings flick before the ex-Burnley striker capitalised on Ben Mee's error to make it 2-2.
The hosts completed the turnaround in the 66th minute when Theo Walcott lifted the ball to the back post and Nathan Redmond volleyed home.
Here are our RATINGS...
9. Josh Brownhill 5.3
Excellent for the first quarter, pressing high, hard and preventing Southampton's more influential figures from getting on the ball. Largely bypassed after the break as the Saints exploited the space in-between the lines. Photo: Clive Brunskill
10. Matej Vydra 7
Lacks composure at times, but battles for every single ball. Didn't give the Southampton defence a moment's peace as he closed the ball down and stretched the play. Took his goal impeccably well when beating Forster from outside the area. Photo: Molly Darlington
11. Chris Wood 7.3
Involved in everything going forward. Had the beating of Vestergaard and Bednarek in the air, which contributed to Burnley's second. Kept his cool to score from the spot - becoming the club's joint leading scorer in the PL - and unfortunate not to get a couple more. Photo: Daniel Leal-Olivas
12. Charlie Taylor 6.3
Partly culpable for Southampton's equaliser when failing to cut out Walker-Peters' pass, but made a number of timely interventions in the second half. Got a touch on Ward-Prowse's effort that came back down off the underside of the crossbar. Photo: Clive Brunskill