Hasenhuttl is preparing for a return match against the high-flying Foxes, little more than two months since coming under severe pressure for the heaviest defeat in Saints' 134-year existence.
The 52-year-old Austrian admits the astonishing result showed his methods were not working and prompted a tactical rethink.
He has since repaid the faith of the south coast club's board by masterminding a fine run of form which has significantly eased his team's relegation fears.
"No, I never thought about walking away," said Hasenhuttl.
"I only thought about hopefully my club trusts me and to show them that I am not at the end with my ideas. Finally the success came back.
"It was an important situation for me. I'm thankful to sit here again. I feel that I have a club who is absolutely behind me.
"We learned a lot, especially that the way we played in that moment, we were absolutely on the wrong path.
"In football, when there comes such an evening with such a result, if you take it as the right chance to turn something then it is a positive evening.
"The chance we took with both hands, everybody, and this is the positive thing."
In addition to the backing of Southampton's hierarchy, Hasenhuttl was also touched to receive supportive messages from fellow managers after Leicester ran riot at a stunned St Mary's in late October.
The drubbing was the biggest home defeat in top-flight history and equalled the heaviest loss of the Premier League era.
Former RB Leipzig coach Hasenhuttl says the humiliation will feature prominently in his preparation for Saturday's visit to the King Power Stadium.
"We will speak about it. We have it definitely as a topic in our preparing for the game," said Hasenhuttl, whose 12th-placed side are five points clear of the relegation zone.
"In what way? I'm not so sure in the moment but we cannot make it unhappen. This result happened and you can take energy out of this result.
"We don't forget this game and to do it better means, for me, to show that we are competitive and I think we can be."