Steven Caldwell's near post header moments before half-time was enough to see off the Saints in a game littered with missed opportunities for the home side.
Strikers Bradley Wright-Phillips and Stern John were the guilty parties as they spurned good opportunities after the break.
Wright-Phillips struck a post from close range while Trinidad and Tobago international John mis-kicked embarrassingly on two occasions when placed just ten yards in front of goal.
The defeat leaves Southampton a point clear of the bottom three and, as their next game is an away trip to West Brom, their season looks as though it will be decided on the final day of the season.
Manager Nigel Pearson admitted his team were not at their best but still thought they did enough to get something out of the game and, true to his word, from start to finish they dominated.
Midfielders Inigo Idiakez and Jason Euell tested the excellent Brian Jensen from long range and the effervescent Andrew Surman made a number of telling breaks forward, but frustratingly they could not match their possession with an opening goal.
That left Burnley free to nick the crucial strike just before the break as first Chris McCann was denied by Richard Wright with a header and, from Robbie Blake's corner Caldwell headed home virtually unmarked for his second goal of the season.
Pearson's side began the second half in a similar fashion to the first with Wright-Phillips clipping the inside of the post after Euell had headed back across goal.
As nerves began to tell, top scorer John then fluffed his lines twice to leave the 21,762 crowd inside St Mary's sensing it was not to be their day.
The passage of the game was perhaps best summed up in the closing minutes when Youssef Safri let fly from 30 yards and, despite the ball looking destined for the top corner, Jensen dived acrobatically across his goal to tip the effort over.