The hosts kept their composure to see out the game, though a blemish to proceedings was an early injury to keeper Fraser Forster.
Here, Sports Mole takes a closer look at the action at St Mary's Stadium.
On target: 3
On target: 3
Was the result fair?
As the stats above show, the two teams mustered a whopping 35 shots between them today. Despite this, each team only managed to get three on target, though that did not detract from the quality of the game. The first half especially was an excellent spectacle, with both sides attacking at will and creating chances. Ultimately, Southampton's professional approach to the second half was the difference, and it is hard to argue that they were not worth the three points, despite Burnley's efforts during the opening half.
Before opening the scoring, there was a period where Southampton were not the better side. Substitute keeper Kelvin Davis made two important stops, but eventually Long's opener settled matters down. Graziano Pelle did not have his best afternoon in front of goal, though Steven Davis was tireless working behind him. It is the Saints defence that is most impressive, with Jose Fonte and Toby Alderweireld looking solid in the centre and Nathaniel Clyne and Bertrand supporting on the flanks. Both full-backs registered assists for the two goals this afternoon.
Sean Dyche's side just could not keep up the intensity with which they played the first half. Scott Arfield and Danny Ings were both lively, while Ashley Barnes and Sam Vokes were willing supporters. George Boyd should have probably had a penalty, and it was a shame for them that they could not convert a chance before half time. The own goal knocked the stuffing out of them, meaning that the remainder of the match simply petered out. They can take heart from a good performance, but if they are going to stay up then they are going to have to score some more goals.
Sports Mole's man of the match
Brought on early to replace the injured Forster, Davis excelled, making two smart stops to deny Ings. He was the Saints' key man at a time when they were looking a little rattled, and his performance deserves credit because it was the difference before the interval.
Shackell takes this award for his own goal, though there was very little that he could have done about it. Bertrand's cross was superb, right into an area which defenders hate, and Shackell could not move quickly enough to avoid diverting it into his own net.
Roger East had an decent game, but the biggest call was one which he perhaps got wrong. Boyd controlled the ball, cut back inside and was shaping to shoot as Fonte wiped out his standing leg. Burnley can feel aggrieved that the decision did not go their way.
With the international break disrupting next weekend's domestic calendar, the Saints will wait until Saturday April 4 to travel to Everton for their next league game.
Things do not get any easier for the Clarets, who welcome Champions League-chasing Tottenham Hotspur to Turf Moor on Easter Sunday.