Hasenhuttl was sacked on Monday after a 4-1 defeat at home to Newcastle which left Saints in the Premier League relegation zone.
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It ended the Austrian's almost four-year spell in charge of the south-coast club, with Austin playing under the coach at the beginning of his tenure.
The Englishman fell out of favour under Hasenhuttl and left his boyhood club eight months into the manager's reign.
Looking back, Austin admits he always had reservations about Hasenhuttl.
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The 33-year-old told talkSPORT Breakfast: "It was all about him.
"He was quick to throw the team under the bus when the team was doing poorly.
"But when they were doing great, it was all about him.
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"When you're being asked if you can get out of a blip and you say, 'No', I don't think any manager can say that. You need to galvanise your team.
"Even if you don't believe it, you need to say you can turn it around."
While Southampton were never truly threatened by relegation during Hasenhuttl's tenure, they never finished in the top-half of the Premier League table.
And Austin questions whether survival can be considered success at St Mary's.
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Austin added: "He was been there for four years and his highest point tally was 52 points, the rest it's been low forties.
"Have they finished in the top-half since he's been there? No.
"Can you say it's a success? You would have to say no - but because he's kept in the Premier League, you could say yes.
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"It depends on what the ambitions of the football club are."
Luton have given manager Nathan Jones permission to talk with Southampton about the vacancy.
Jones has guided the Hatters into Championship play-off contention, but Austin says his disappointing 11-month spell at Stoke in 2019 casts doubt on his credentials.
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Austin continued: "I understand he's done a great job at Luton, but when he went to Stoke, it didn't work out.
"Stoke might've been all over the place, but he failed with big names.
"He has a bigger job on his hands keeping Southampton in the Premier League than getting Luton in the play-offs."