Burnley had the upper hand for much of the first hour, with McCarthy saving from Aaron Lennon and Jeff Hendrick. Southampton's Jack Stephens also denied Johann Berg Gudmundsson with a superbly timed challenge.
Southampton pushed for a goal after the introduction of new signings Danny Ings and Mohamed Elyounoussi in the 56th minute.
Elyounoussi had a shot headed off the line by Ashley Westwood before Hart, on his PL debut for Burnley, saved from Stephens and Mario Lemina. Ings also came close with two efforts.
Southampton: 18 Aug v Everton (A)
Burnley: 19 Aug v Watford (H)