Saints splashed out Â£4million to land striker Austin from QPR on Saturday morning in a bid to pep up their misfiring attack.
And just a couple of hours later Austin watched from the stands as James Ward-Prowse scored twice before Dusan Tadic wrapped up a comfortable victory.
England Under-21 midfielder Ward-Prowse was an unlikely goal hero, having previously scored just once in the Barclays Premier League
But he doubled that tally with just five minutes on the clock after Craig Dawson fouled Sadio Mane on the corner of the penalty area
Ward-Prowse curled the free-kick up and over the wall, his effort beating Baggies goalkeeper Boaz Myhill far too easily as it flew into the top corner
West Brom, who came from behind twice to draw at Chelsea in midweek, had chances to equalise but Craig Gardner headed James McClean's cross wide and blazed another chance over
Yet Saints looked dangerous every time they attacked and Myhill kept out a Shane Long drive, although the keeper was grateful to Claudio Yacob for hacking the ball clear after making a mess of a corner.
And in the 33rd minute the hosts' pressure told with Dawson again the guilty party, although the defender was perhaps unlucky that his collision with Matt Targett in the area was adjudged to be a foul.
Nevertheless, with regular penalty taker Tadic on the bench, Ward-Prowse did the honours with a coolly taken spot-kick which sent Myhill the wrong way.
Saints would have been home and hosed by half-time but for a goal-line clearance by McClean from Targett's low drive.
Frustrated Albion boss Tony Pulis made a double change at the break, sending on Chris Brunt and Salomon Rondon, but Southampton continued to create the better chances.
Austin will have been licking his lips at the prospect of getting on the end of some of them, not least when Long sent Mane clean through only for the frontman to shank his shot across Myhill and wide of the far post.
Tadic was introduced from the bench shortly after the hour mark and got on the scoresheet just eight minutes later, finishing with an angled snap-shot after he was played into the area by Steven Davis.
With victory assured, Southampton fans began serenading their former hero Rickie Lambert, who must have been wondering why he ever left the club he could not stop scoring for just to end up sat on the bench at West Brom.
Saints will be hoping they have just snapped up a new goalscoring talisman in the shape of Austin.
TWEET OF THE MATCH
"Good first half, @SouthamptonFC looked in control at the back and dangerous up front, great goal @Prowsey16!" - former defender and now Southampton Under-21s assistant Radhi Jaidi's half-time assessment was equally relevant at full-time
Fraser Forster: 6
Cedric Soares: 6
Jose Fonte: 7
Virgil van Dijk: 7
Ryan Bertrand: 6
Matt Targett: 7
Steven Davis: 8
James Ward-Prowse: 7
Victor Wanyama: 6
Sadio Mane: 6
Shane Long: 7
Dusan Tadic: 7
Oriol Romeu: 6
Cuco Martina: 6
Boaz Myhill: 5
Craig Dawson: 5
Jonas Olsson: 5
Gareth McAuley: 5
Jonny Evans: 6
Claudio Yacob: 5
James McClean: 6
Darren Fletcher: 5
Craig Gardner: 6
Stephane Sessegnon: 5
Victor Anichebe: 5
Chris Brunt: 6
Salomon Rondon: 5
Saido Berahinho: 5
Steven Davis - The midfielder once again pulled the strings for Southampton with a no-fuss display
Nice pass to Dusan Tadic to set up the killer third goal, capping another impressive performance.
MOMENT OF THE MATCH
James Ward-Prowse had only ever scored one Premier League goal before stepping up to curl a superb free-kick home
A certain Gareth Bale, formerly of this parish, would have been proud of that one.
VIEW FROM THE BENCH
Saints boss Ronald Koeman has taken to playing with three at the back, with Matt Targett and Cedric Soares as wing-backs
Albion chief Tony Pulis shuffled his pack but there was still no place for ex-Saints hero Rickie Lambert, with Victor Anichebe preferred up front.
MOAN OF THE MATCH
West Brom were truly pitiful and, very unlike a Tony Pulis side, they showed little appetite for the battle
They went toe-to-toe with Chelsea in midweek and deservedly earned a point, but here they went down with a whimper.
Man United v Southampton (Barclays Premier League, January 23)
Bristol City v West Brom (FA Cup third round replay, January 19)
Source : PA