Jones takes over from Ralph Hasenhuttl at St Mary's after the Austrian manager was sacked on Monday following their 4-1 home defeat to Newcastle United last weekend.
Luton Town granted permission for Jones to speak to Southampton earlier this week, and the Saints have moved quickly to appoint the 49-year-old on a deal which runs until 2026.
"I'm really proud to be given this opportunity. I know a lot about the club from back in the days of The Dell, to coming here to St Mary's, and it's a wonderful football club," the Welshman told the club's official website.
"A lot of my family are Southampton fans, which doesn't half help, and I feel really, really proud to be given the opportunity, and I'm really looking forward to getting started.
"Obviously, I wanted to manage in the Premier League, I've dreamt of that since I've become a coach or a manager, but this club in particular - because of how it's run, because of the structure, because of how they look deeper than just results - really appeals to me."
Jones will be joined at Southampton by first-team coaches Chris Cohen and Alan Sheehan, who also make the move from Luton.
The former Stoke City boss was named the Championship manager of the season for 2021-22, having steered Luton into the playoffs, where they were beaten in the semi-finals.