Welshman Connor Roberts breathed fire into the home side’s evening with his first goal for the club in the 12th minute.
The right back opened up his body after collecting Josh Brownhill’s pass and whipped the ball into the far corner with his left boot.
Saints stopper Fraser Forster denied Wout Weghorst and Jay Rodriguez twice, while the Clarets’ Dutch forward missed out on his first goal on home soil when his header came back off the post.
Midfielder Oriol Romeu should have done much better for the visitors when failing to hit the target with a couple of free headers from James Ward-Prowse’s dangerous deliveries.
Fraser then denied Dwight McNeil at the foot of his post, but the 34-year-old ex-Celtic goalkeeper could only watch as Nathan Collins doubled the home side’s lead.
There was just a minute remaining in the first half when the Republic of Ireland international climbed above Ward-Prowse to guide Brownhill’s set-piece into the corner.
Nick Pope made a big save to deny substitute Che Adams in the second half when the Scot met Nathan Redmond’s pass with a first-time finish while Charlie Taylor did impeccably well to get across to deny the striker what seemed a certain goal soon after.
However, the Clarets preserved their clean sheet to see the game out and move within a point of 17th place Everton.
Here are the ratings.
5. Dwight McNeil 8.5
Back to his brilliant best. Thrived in his role as an inverted winger, driving at players, carving open space, combining and delivering. Worked hard, pressed high, played with a smile on his face and stardust in his boots. His decoy run took Armstrong away to open up the space for Roberts' effort, he laid the ball on for Weghorst in two big moments, tested Forster at the foot of his post and was a whisker away from scoring when cutting inside late on. Photo: Clive Brunskill
6. Maxwel Cornet 6.5
Yet to convince that the left hand side of midfield is the most suitable position for him. Works hard for the ball and always a willing runner, which enables the Clarets to push high up the pitch, but couldn't effect the game in possession. Looked as though he was stuck in treacle at times when the ball was played in to him. Replaced by Lennon (66). Photo: Clive Brunskill
7. Josh Brownhill 8
Trebled his number of assists for the season when claiming two against Southampton. Laid the ball into Roberts for the opener and then supplied the corner for Collins to double the advantage. Broke the play up well, made blocks at his own byline when racing back to aid his defence, and knitted play together nicely. It's a cliche, but the midfielder covered every blade of grass. Photo: Clive Brunskill
8. Jack Cork 8.5
The midfielder just makes a difference when he's in the side. Calm and controlled in possession, energetic off the ball, shields his defence efficiently and effectively, and his patrolling of midfield provides Brownhill with a licence to advance. Did remarkably well to prevent Ward-Prowse from dictating play. An excellent evening against his former club. Photo: OLI SCARFF