James Ward-Prowse's penalty and Shane Long's first goal since April 2018 earned the battling 10-man Saints a 2-1 victory at Leicester.
Long scored in first-half injury time just two minutes after Yan Valery was sent off for two bookable offences.
Wilfred Ndidi's consolation failed to rally Leicester as Hasenhuttl lifted the Saints out of the Premier League relegation zone.
He said: "Every win would be very significant, it's an important step but only one. The decision won't be before May, it's so close and there are so many good teams.
"We showed our potential to win the games, that's what I've demanded from the team.
"I'm really proud of this fantastic team and the commitment we showed. It was a very long time to go after the red card but we were not frightened.
"The first (Leicester) goal was a little too early and I thought it would be a little difficult to take something. I have a feeling the players enjoyed it."
Hasenhuttl has won three of his eight games – the same number in the league predecessor Mark Hughes managed during his 10-month St Mary's tenure.
Ward-Prowse netted from the spot after 11 minutes when Nampalys Mendy fouled Long and the midfielder gifted Long the ball to add a second.
"Shane is the best example of what we demand, hard working, good defending and if you get the chance to kill it, do it," added Hasenhuttl.
"I know he hasn't got the best finishing record but I know previously he scored 10 to 12 times. I want to bring him back into this shape."
It came after Valery walked for tugging back Marc Albrighton having already been cautioned for a spat with Ben Chilwell.
Ndidi bundled in mid-way through the second half but Leicester lacked inspiration all game after last week's FA Cup shock of losing 2-1 to Newport.
The Foxes were booed off with at least one 'Puel Out' sign spotted but the Frenchman will ignore the critics.
Puel said: "The fans are disappointed like us and it's a normal thing, it's not my concern, I have to keep my focus and concentrate on the team.
"It's a big disappointment, we wanted a good response after our Cup game. The players were focused, perhaps a little nervous, and we showed this in the first half.
"We wanted to perform, we wanted to have a reaction and our concentration was perhaps too much. It was a difficult scenario.
"We didn't manage the first half well. It would have been good to have come back without the second goal but it was harsh and it was a bad moment.
"We played with desire and intensity in the second half but we didn't perform enough."