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.
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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.