Burnley took the lead in bizarre circumstances after just 93 seconds when Danny Ings left Ashley Westwood's corner at the near post and Alex McCarthy, who was standing behind his line, was unable to keep it out.
Ings atoned for that error by firing his 15th league goal of the season into the bottom-right corner from the edge of the box in the 18th minute.
Jack Stephens headed a cross from Sofiane Boufal, who went off injured in the first half, against the crossbar before the break and Burnley regained the lead on the hour mark, Vydra controlling Jeff Hendrick's long diagonal pass brilliantly before drilling home.
Southampton appealed for a penalty after Stephens' cross hit Ben Mee's arm, but referee Simon Hooper let play continue and VAR (video assistant referee) backed up his decision.
Nick Pope did well to deny Pierre-Emile Hojbjerg in the 90th minute and Sean Dyche's men hung on to make it four top-flight matches without defeat, while the Saints have won just one of their past five across all competitions.