Saints won 4-0 at Fratton Park, their first win at the venue since 1984, to book their place in the Carabao Cup fourth-round.
Speaking after the game, an elated Hasenhuttl said: "We will not forget this evening, and the fans too.
"It was a fantastic atmosphere until the end and to score four goals here is fantastic.
"It is more than only reaching the next round. It is about reaching the hearts of the fans.
"I am very happy for them (the Saints fans).
"I knew it was massive game for them and I think we really didn't start well in this game but the players knew they had to fight for every inch and after the 1-0 it was a much better game for our side.
"After the 2-0 I was not sure if this game is done for us so you never know if they make the 1-2 then suddenly everything is alive again and it gets difficult for you. At 3-0 you know it is done.
"It was a hard job to do but the guys did it well."