Hasenhuttl replaced Mark Hughes after the Welshman was sacked in December, and kicked off his tenure with a 1-0 defeat away to Cardiff.
It is the Saints' home form, however, that is most worrying.
The club last won at St. Mary's in April, and their previous home league victory before that was all the way back in November 2017.
But the former RB Leipzig boss is looking to get the fans onside straight away with the sending of free drinks vouchers to season ticket holders.
"You have to build up your credit," he told reporters ahead of Sunday's test.
"It's like an account - if you always take money off sometimes it's empty. Now we have to put money on the account.
"We try to invest something and show them that we are working harder and put more in games."
Hasenhuttl also addressed long-suffering Saints fans in a letter, where he promised to give his all in order to steer the side away from relegation.
"If they go with me on this, then they will find a very rewarding journey," he wrote.
"The other people I want to make sure I bring with me are you, the fans."
Southampton are currently 19th in the Premier League, with just nine points from their opening 16 matches and ahead of bottom dwellers Fulham only on goal difference.
Their only victory to date came on the road, with fellow relegation strugglers Crystal Palace dispatched 2-0 at Selhurst Park.