The 27-year-old returned to St Mary's over the summer after a two-year loan spell with Scottish giants Celtic, where he scored 24 goals and provided 14 assists in 67 appearances across all competitions.
Elyounoussi is now considered a first-team regular under Hasenhuttl and he is pleased to be playing regularly for the Saints in the Premier League.
"When I got the opportunity when the Saints came in and gave me an offer, it was a huge dream," the Norway international said in Southampton's most recent matchday program.
"It was a big move, I knew that Southampton has followed me for a few years, especially during my last season in Basel. I think they were there almost every game, so that made me comfortable to come here.
"It didn't work out as we thought at the beginning, I needed time to adapt, I had a few injuries and things just didn't go my way, really. I needed time to get better, to get game time and find myself again and get some confidence.
"I am really happy I went through something like that and came back to show a better side of myself. I don't know many players that have been sent out on two loan seasons and then come back to make it in the team. It shows the belief that the club and the manager have in me to give me a second chance, I'm just glad that I took it."
Despite reported interest from Arsenal and Leicester City, Elyounoussi added: "I remember, even though I didn't play much, I had a really good time at Southampton, both inside and outside. My family enjoy it here. We're settled down now and my daughter has just found a nursery, so we're back to where we belong."
Elyounoussi has made 13 appearances for Southampton across all competitions so far this campaign, scoring four goals and registering two assists. body check tags ::