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.
1. Nick Pope 6.5
Looked to be wrong-footed when Armstrong pulled a goal back for the Saints, but made a couple of tremendous saves in the second half to keep the Clarets in the game. The England international did incredibly well to keep out Armstrong and Ings.
Photo: Jon Super
2. Matt Lowton 5.3
Just couldn't settle into the game. Gave the ball away cheaply on too many occasions in the first half, with the Saints capitalising on one such error. Slow to get out to Redmond for Southampton's winner.
Photo: Jon Super
3. Erik Pieters 6.4
Enjoyed a vibrant 33 minutes before being forced off through injury. Took up some good attacking positions down the left hand side, finding plenty of space, and drew the foul from Walker-Peters to win the penalty.
Photo: Alex Pantling
4. James Tarkowski 6.4
Seemed distracted when Southampton found a way back into the game, which afforded Armstrong far too much time to finish. Much of the defending seemed desperate as they failed to address the movement of Southampton's front two.
Photo: Paul Ellis