All four goals, including two penalty kicks, arrived in an explosive second period on the south coast, but it was the Saints who did enough to pick up the win which takes them into eighth.
Burnley did break their opponents' watertight defence by bagging a goal in the final quarter of the match - the first Southampton have conceded in 612 minutes - yet the damage had already been done as their wait for an away point goes on.
Charlie Austin proved to be the hero with two goals against his former side, while Tom Heaton also did his reputation no harm with one eye on England's next squad after making a string of impressive first-half saves.
It took seven minutes for Southampton to create their first opening of the match after a cross from the left was met by the head of Austin, only for Heaton to produce the first - and best - of his many stops.
The Englishman got across superbly to palm the ball aside, and a more straightforward stop was to come a few minutes later as Dusan Tadic blasted the ball straight at him - the second of 14 shots on goal for the home side.
Both teams lost a player to injury in the opening 45 minutes - Matt Targett, in for Ryan Bertrand who is himself on the sidelines, hobbled off to make way for Premier League debutant Sam McQueen, while Burnley had to take off key man Steven Defour.
The Clarets failed to register an attempt on target in the first half - their best chance falling to Sam Vokes midway through when the childhood Southampton fan volleyed Stephen Ward's cross wide of the target.
The big moments in the match were continually involving Heaton, who denied Tadic 35 minutes in with his upper body when the creative midfielder was played through in bizarre circumstances by Fraser Forster's punt upfield.
That came moments after Johann Berg Gudmundsson saw his appeals for a penalty under a challenge from Virgil van Dijk turned down by Mike Dean, but that proved to be a rare attacking move from the visitors as Southampton continued to knock on the door.
Tricky winger Redmond twice used his quick footwork to turn an opponent inside out and test Heaton, with the England back-up keeper - a doubt for today's game after withdrawing from international duty - coming out on top on both occasions.
Heaton, who has now played 138 games in succession for Burnley since joining more than three years ago, was to be tested once more prior to the break by a long-range Van Dijk drive, which was never going to find a way past the in-form stopper.
It took just seven minutes of the second half for Southampton to finally break the deadlock, though - Austin getting on the end of Van Dijk's back-post header to help the ball over the line at the second time of asking, making it three in three against his former club at that point.
The Saints were well on their way to three points when Redmond doubled his side's tally on the hour, peeling off his marker and blasting the ball through Heaton, who had denied Tadic to concede the initial corner.
It was three soon after thanks to a second Austin goal of the day, with the ex-Burnley man converting from the penalty spot after Gudmundsson brought down the lively McQueen inside the area.
Burnley were given a slight lifeline with a little over 15 minutes to play as the goals kept on coming on the south coast, this time when Tadic pushed Ben Mee which allowed Vokes to himself convert from the spot.
That goal would not alter the end result but it did bring an end to Southampton's run of clean sheets, which stood at six matches heading into this clash, while also getting the Clarets off the mark in terms of their first away goal this term.
Southampton, now without defeat in their last nine home games in the Premier League, were put under a little pressure in the remainder but it was they who came closest to another goal as sub James Ward-Prowse and Jose Fonte both came close late on.