Austin was left unmarked when fellow substitute James Ward-Prowse crossed the ball in from a free-kick. Austin rose highest and powerfully headed the ball past David De Gea into the bottom corner.
Chances were limited in the game with Jesse Lingard's deflected effort and Adnan Januzaj's late strike both going wide. Southampton also went close through Sadio Mane and a Victor Wanyama header with Oriol Romeu close to making it 2-0 late on.
Charlie Austin had to watch from the bench for 80 minutes with Koeman opting to leave the striker on the bench with Shane Long leading the line with a tireless performance. The Saints boss made two changes with Jordy Clasie and Dusan Tadic coming in for an unfit Steven Davis and last week's double goalscorer Ward-Prowse.
Saints were the better team in the first half which came out of their good defensive organisation and the dominance of Wanyama and a fantastic performance from Clasie in midfield.
The best chance of the first half fell to Southampton with Mane and Tadic linking up well with a one-two. Mane played the ball to Tadic and the Serb played in a lovely ball behind the United defence for Mane. The forward's first touch bounced up onto his thigh and De Gea was able to gather the ball with Mane one-on-one with the United 'keeper.
On the brink of half-time, Fraser Forster watched a deflected Lingard effort bounce wide. The young winger's shot deflected off his teammate Ander Herrera and just went wide of Forster's goal.
United left the field at half time to boos from the home crowd after their lacklustre opening 45 minutes. Mata's introduction at the beginning of the second half seemed to improve United's play with van Gaal's team playing with greater urgency and tempo but Southampton slowly came back into the game.
Just after the hour mark, Wanyama should have done better from a Ryan Bertrand corner. The Kenyan peeled off Wayne Rooney but could only head wide from close range.
The match-winning moment came three minutes from full time. Ward-Prowse delivered a free kick near to the corner flag into the box and it was met by Austin. The debutant was unmarked and he powered the ball past De Gea.
It could have been a more comfortable last few minutes for Southampton when Romeu went close in stoppage time. The Spaniard picked the ball up off Long and fired a low shot at De Gea but Romeu's compatriot parried the ball well.
United piled on the pressure in the final few minutes. A long ball forward was knocked down and it fell to Januzaj. The Belgian met the ball first time with a powerful effort which went just wide of the Southampton goal.
The full time whistle was met with boos from the home crowd. Koeman has now done the double over van Gaal and United at Old Trafford and it was the perfect performance from his side and a perfect way for Austin to start his Southampton career.
Saints now move to eighth in the table after securing their third consecutive win and clean sheet and are now just four points off a Europa League spot.