Southampton thrashed fierce rivals Portsmouth 4-0 on Tuesday to seal their place in the Carabao Cup fourth-round.
Danny Ings scored twice while there were also goals for Cedric Soares and Nathan Redmond.
It was Saints' first win at Fratton Park since 1984.
"This is more than only reaching the next round," said Hasenhuttl.
"This is about reaching the hearts of the fans. I am very happy for them.
"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."