Hart, who made his Burnley debut in Thursday's 0-0 draw against Istanbul Basaksehir in Europa League qualifying, made a couple of outstanding saves while former Clarets favourite Danny Ings made a similarly eye-catching debut for Southampton after his loan move from Liverpool.
"I push myself to the edge in all I do. I’m really pleased to be at this club and to be here permanently," Hart said. "I've walked into a nest of top keepers and I'll have to be on my toes if I want to keep the shirt."
Burnley had the best of the first half, firing on sight at Alex McCarthy in the Saints goal with the goalkeeper doing well to prevent the visitors going in ahead.
The pick of his saves was denying former England international Aaron Lennon from point-blank range, before he got down well to save Ireland international Jeff Hendrick's header at the end of the first half.
McCarthy was beaten once but former Saints midfielder Jack Cork had his goal ruled out for a marginal offside call.
Hart had less to do in the first half but proved equal to Mario Lemina's header, tipping it over the bar.
Burnley started the second half as they had finished it on the front foot and Icelandic international Johann Berg Gudmundsson and Stephen Ward both went close to breaking the deadlock.
Saints manager Mark Hughes decided he needed to add some zip into his lethargic team and sent on two of his close-season signings, including Ings.
However, it was Burnley defender Ben Mee who threatened Hart's goal when his miscued headed clearance looped towards the top corner, only for Ashley Westwood to be alert and head it off the line.
Hart then produced a superb save to deny Jack Stephens, who had been picked out by Ings.
Ings -- who scored 43 goals for Burnley before earning his move to Liverpool, which floundered due to injury -- went close to opening his account with a header from a corner but it cleared the bar.