Alex McCarthy was equal to a succession of first-half Burnley chances, while the Saints finished the stronger but could not find a breakthrough to pinch all three points.
However, it was one of their own former players in the Burnley ranks who found the net on seven minutes. The Clarets recycled a corner to Jack Cork who finished confidently into the bottom corner from 12 yards, only for the offside flag to go up.
The visitors continued to have the better of the opening stages and needed McCarthy, preferred to new £12m signing Angus Gunn in goal, to rescue them when Aaron Lennon ran onto a botched clearance from Wesley Hoedt to volley powerfully from close range.
McCarthy may have been beaten a couple of moments later when Johan Berg Gudmundsson whipped a 25-yard free kick over the wall, which dropped agonisingly onto the roof of the net.
Crosses into the box were Southampton's best source of opportunities, and they had a new weapon from set pieces in 6ft 6in defender Jannik Vestergaard, who headed over the bar from a 41st-minute corner.
Burnley should have gone into the break in front but James Tarkowski could not direct his close-range header past McCarthy, who also saved from Chris Wood from the resultant corner although the striker did not generate much power on his header.
Vestergaard make an impressive debut for the hosts, crucially intercepting a Matt Lowton cross from which Wood was lining up a simple tap-in. The Dane also deflected Jeff Hendrick's shot wide as the Clarets started the second period as brightly as they did the first.
The double introduction of new recruits Ings and Mohamed Elyounoussi gave the Saints some attacking impetus but it was actually a Burnley player who nearly gave them the lead when Ben Mee's header was cleared off the line by Ashley Westwood.
Ings looked particularly lively against his former club, curling a 25-yard shot which had Hart scrambling narrowly over the crossbar. The on-loan Liverpool forward also teed up Jack Stephens who ran onto the cross, but was denied by a well-positioned Hart at the near post.
Hart was assured all afternoon though and held another Lemina header on 74 minutes. Southampton came again from another corner when Ings got away from marker Lennon, only to direct his effort high of the goal.
In the end it was a match in which the defences - and goalkeepers in particular - came out on top.
SOUTHAMPTON (3-5-1-1): McCarthy; Cedric (Elyounessi, 56'), Stephens, Vestergaard, Hoedt, Bertrand; Romeu, Lemina, Armstrong (Ings, 56'); Redmond; Austin (Gabbiadini, 72')
BURNLEY (4-5-1): Hart; Lowton, Tarkowski, Mee, Ward; Lennon, Cork, Hendrick (Barnes, 76'), Westwood, Gudmundsson (Taylor, 87'); Wood (Vokes, 73')