The home side almost went in front as early as the second minute when striker Robbie Blake crashed a shot against the underside of the crossbar.
But the Clarets didn't have long to wait before they went in front when Wade Elliott's low cross into the box was deflected by the left leg of veteran central defender Chris Perry beyond goalkeeper Kelvin Davis.
Burnley increased their lead just three minutes later when Blake's corner was headed towards goal by Steven Caldwell but fumbled by Davis which allowed Joey Gudjonsson to score from close range.
Things got even better for Owen Coyle's side when they scored for a third time with only 11 minutes on the clock when, after good work by Martin Paterson, Elliott's cross to the near post was turned in by Gudjonsson.
Burnley should have added to their tally on 19 minutes when a shot from Chris McCann was parried by Davis but striker Paterson could only hit the post with the rebound.
Southampton's only goal threat in the opening half came nine minutes before the interval when David McGoldrick headed wide of the target.
But it was a different story after the re-start with Southampton pulling a goal back just three minutes after the interval when Bradley Wright-Phillips pulled the ball back from the by-line and left-back Rudi Skacel crashed his shot high into the net.
Southampton pulled a further goal back on the hour mark when Jason Euell played the ball into the path of Andrew Surman who gave Jensen no chance.