The 18-year-old midfielder has moved to St Mary's for a fee believed to be in the region of £12m plus add-ons, and the Citizens are understood to have a £40m buy-back clause.
Speaking to the club's official website on his arrival, Lavia said: "I'm really delighted to join this club. I think this club has a really good culture with young players, so I'm really happy to be here.
"When I heard of the interest of Southampton I was buzzing because there's an opportunity for young players to develop as much as possible and I'm very excited."
Saints boss Hasenhuttl added: "Romeo is an extremely talented young player at the start of his career who is open-minded and wants to learn and develop his game with us.
"We are very excited to sign him ahead of many other interested clubs. It proves that we have a positive reputation for improving young players, which we have worked hard to build up over the years.
"Romeo is coming from one of the best teams in the world, learning from the best manager and the best players. Now it is up to him to show us why he deserves to play in the Premier League, as we hope to help take his game to the next level."
Lavia joined Man City's academy from Anderlecht in the summer of 2020 and developed into a key player at Under-18 and Under-23 level.
Last season, the midfielder made 27 appearances for City's Under-23s side who won the Premier League 2 title, and he also featured twice for the senior team under Pep Guardiola, including his debut against Wycombe Wanderers in the EFL Cup in September last year.
Lavia, a Belgium Under-21 international, will now be looking to impress Hasenhuttl in pre-season training this summer and will bid to make his Premier League debut at some stage during the 2022-23 campaign.