Ashley Westwood scored from a bizarre situation when his corner kicked for the net after Danny Ings failed to clear the ball after just 100 seconds.
Ings, however, redeemed himself on 18 minutes after turning the Burnley defence before hammering a right-footed shot into the goal, tallying his 15th league goal of the season and 18th in all competitions.
Ings sits just six goals behind James Beattie’s top-flight mark of 24 for Southampton, which came during the 2002-03 campaign.
Southampton played the better of the two sides, but it was Burnley’s Matej Vydra who broke the deadlock on the hour mark with a thumping effort.
Pierre Emile Hojbjerg had Saints’ best chance to draw level again, but he could only dig the ball out of his feet long enough to fire straight into Nick Pope nearing full-time.
Southampton: McCarthy, Bertrand, Vestergaard, Stephens, Walker-Peters (Obafemi 73), Boufal (Djenepo 39), Hojbjerg, Ward-Prowse, Armstrong, Ings, Long (Long 86)
Unused subs: Gunn (GK), Bednarek, Romeu, Smallbone
Goals: Ings (18)
Bookings: Vestergaard (17), Obafemi (88)
Burnley: Pope, Tayor, Mee, Tarkowski, Bardsley, McNeil, Westwood, Cork, Hendrick, Rodriguez, Wood (Vydra 22)
Unused subs: Hart (GK), Lennon, Pieters, Long, Brady, Brownhill
Goals: Westwood (2), Vydra (60)
Bookings: Hendrick (26), Mee (44)