Austin, signed from Queens Park Rangers a week ago, emphatically headed home James Ward-Prowse's 87th-minute free kick to settle a game of few chances.
The win was Saints' third in a row – and also means they have won at Old Trafford for two seasons in succession.
Boss Ronald Koeman had made two changes to the side that started last weekend’s win over West Bromwich Albion, one of which was enforced by an injury to Steven Davis. He was replaced by Dušan Tadić, who scored from the bench against the Baggies, while Jordy Clasie was preferred to Ward-Prowse.
United made just one alteration from the previous weekend’s morale-boosting win over Liverpool, as youngster Cameron Borthwick-Jackson stepped in for the injured Ashley Young. Former Saint Morgan Schneiderlin started in holding midfield.
Koeman stuck with the 5-3-2 system that had served Saints well in their back-to-back home wins, with United boss and fellow Dutchman Louis van Gaal choosing to adopt the same tactic. Interestingly, it was Koeman’s use of the formation with previous club Feyenoord that was credited as van Gaal’s inspiration for deploying it during the Netherlands’ impressive showing at the 2014 World Cup.
There was nothing impressive about the first half-hour of this encounter, though, as both sides cancelled each-other out.
United did sent a trio of speculative shots towards the Saints goal, with Anthony Martial and Wayne Rooney driving efforts wide either side of Daley Blind having one held by Fraser Forster.
Saints survived a couple of injury scares, with Victor Wanyama requiring treatment after being caught by Marouane Fellaini’s flailing arm, while Clasie came out worse-off in a 50-50 challenge with the same player a few minutes later.
With Clasie prone on the Old Trafford turf, United countered with pace – but Ryan Bertrand did superbly to not only keep up with Martial but also win a free-kick after being tugged back by the Frenchman.
Forster’s first real test of the afternoon came, ironically, after a lapse of concentration by one of his teammates. Virgil van Dijk’s pass back to his goalkeeper lacked power, but Forster came storming off his line to boot the ball clear before Martial could intercept.
Saints’ only chance of the first period came on 37 minutes as Sadio Mané played a smart one-two with Tadić only to see the ball bounce of his thigh as he latched onto the return pass a few yards out, with David de Gea duly gathering.
Having previously seen their side score just 12 league goals at home this season, United’s fans grew frustrated as the half whiled away. Seeing a goal kick awarded after Jesse Lingard’s shot deflected wide off Bertrand only served to compound their annoyance at the end of a drab first period.
Van Gaal acted swiftly at the break, replacing the lumbering Fellaini with Juan Mata, with Ander Herrera – who had made little impression in the number-ten position – now sitting next to Schneiderlin in defensive midfield.
United immediately looked more dangerous, with Mata at the centre of their attacking endeavours. The Spaniard was close to teeing up Martial to open the scoring on 53 minutes after wriggling free of Bertrand, but van Dijk produced a vital challenge before the ball eventually ricocheted to the safety of Forster’s grasp.
When another United attack broke down in the Saints boss, Mané led a rapid counter that ended with Shane Long and Matteo Darmian colliding as they attempted to reach the Senegalese’s hanging cross. Long was able to continue, but Darmian had to be replaced by Paddy McNair.
The stoppage was well-timed from Saints’ point of view, with the wind seemingly taken out of United’s sails just as they gathered momentum.
The visitors soon went close from two corners in quick succession. First, Long’s header from Clasie’s delivery deflected wide, then Wanyama nodded off-target when Bertrand swung the ball over from the opposite side.
With the game gathering pace, Koeman made a double substitution on 67 minutes. Off came Clasie and Tadić, with Oriol Romeu and Ward-Prowse drafted on. It was Saints' third and final change of the game, though, that was met with the most excitement from the travelling fans, as Austin – signed a week ago – replaced Mané for the last ten minutes.
And the celebrations were even louder eight minutes later. After Austin had been fouled on the right flank, Ward-Prowse hung over a teasing delivery for the debutant to delicately nod beyond de Gea.
Another of Saints' substitutes, Romeu, almost wrapped things up in the first of five minutes' added time, but his close-range shot was parried. Adnan Januzaj, on for Cameron Borthwick-Jackson, then flashed an effort wide in the final seconds.
Having waited since 1988 for a victory at Old Trafford, Saints have now won twice there in little over a year – an achievement that the travelling fans duly toasted come full time.
Manchester United (5-2-1-2): David de Gea; Jesse Lingard, Matteo Darmian (Paddy McNair 60), Chris Smalling, Daley Blind, Cameron Northwick-Jackson (Adnan Januzaj 86); Marouane Fellaini (Juan Mata 46), Morgan Schneiderlin; Ander Herrera; Wayne Rooney (c), Anthony Martial. Unused substitutes: Sergio Romero (GK), Memphis, Guillermo Varela, Andreas Pereira.
Southampton (5-2-1-2): Fraser Forster; Cédric, Virgil van Dijk, José Fonte (c), Ryan Bertrand, Matt Targett; Victor Wanyama, Jordy Clasie (Oriol Romeu 67); Dušan Tadić (James Ward-Prowse 67); Shane Long, Sadio Mané (Charlie Austin 79). Unused substitutes: Maarten Stekelenburg (GK), Maya Yoshida, Cuco Martina, Juanmi.
Goal: Austin (87)
Referee: Michael Jones