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.
1. Nick Pope 7.5
The England international didn't have a great deal to do in the first half, but was made to work for his clean sheet after the break. After turning a deflected cross over the crossbar, he made a fine save to keep out substitute Adams' first time effort from a Redmond pass. Claimed a number of crosses to ease the pressure as the Saints searched for a way back into the game. Photo: George Wood
2. Connor Roberts 8
The Welshman's first goal for the club — which he claimed to have envisaged in his dreams the night before — was one to savour. A simply sumptuous finish on his weaker foot, which comfortably beat an in-form goalkeeper in Forster. Always showed for the ball, effective when combining with McNeil, and added a solid defensive display. Photo: Catherine Ivill
3. Charlie Taylor 8.5
The full back enjoyed a competitive battle with Walker-Peters, who pushed high and wide, particularly in the first half. Given a licence to roam when the Clarets were on top and had the Saints pinned back. He made a quite brilliant block to prevent Adams from scoring what seemed a certain goal in the second half. Photo: Catherine Ivill
4. James Tarkowski 8
Aside from a single moment of alarm in the second half, Tarkowski made another Premier League game look like a stroll in the park. Caught underneath Redmond's cross prior to Taylor's block on Adams, but didn't put a foot wrong otherwise. Won practically everything that was there to be won in the air and wasn't tested by Armando Broja. Photo: OLI SCARFF