The 51-year-old Dutchman, who left Feyenoord at the end of last season, has signed a three-year deal with his brother, Erwin, appointed as his assistant.
Koeman's physical coach at Feyenoord, Jan Kluitenberg, will also take up the role of assistant first team and physical coach at St Mary's.
Koeman said: "I am thrilled to be the manager of Southampton. From my very first meeting with Les Reed, I could see clearly that this was the right club for me.
"We share the same philosophy of football, and the same belief in how to develop the club. Southampton have a talented first-team squad, a world-class academy and the infrastructure to sustain success in the Premier League.
"The opportunity to continue the development of this famous football club was simply too good to turn down. I cannot wait to get started."
Saints executive director Reed told Sky Sports News: “Ronald has got the right profile for us. He is used to working with and bringing young players into top level football.
“He is a proven coach in terms of improving players and developing youngsters. He’s also been very successful as player and a coach, so he fits our need to take the club to the next level.
“I think he’ll adapt very easily. Everyone knows a lot about the Premier League now and he’s studied it for a long time, and understands it.
“There will be a few things he needs to adjust to but it’s about the qualities and profile of the man rather than where he has coached. He’s coached all over the world and adapted, so why not here?"
Sky Sports revealed over the weekend that Koeman was the club’s preferred choice after a wide search for Mauricio Pochettino’s successor, who left to join Premier League rivals Tottenham last month.
The duo boast extensive managerial experience, with Ronald having bossed Ajax, Benfica, PSV Eindhoven and Valencia before taking over at Feyenoord in 2011.
Koeman never won the Eredivisie with Feyenoord but did so twice with Ajax in 2002 and 2004, and again with PSV in 2007. He lifted the Copa del Rey with Valencia in 2008.
Erwin, meanwhile, managed Feyenoord between 2005 and 2007, as well as Utrecht, the Hungarian national side and RKC Waalwijk, whom he left this summer, but he has no managerial honours to his name.
Their first assignment will be to try and hold together a squad that has already lost star striker Rickie Lambert to Liverpool and been the subject of extensive transfer speculation following the popular Pochettino's departure.
Left-back Luke Shaw, as well as fellow England international Adam Lallana, have both been linked with moves away from the south coast, while Jay Rodriguez, among others, is also being targeted by other Premier League clubs.
Southampton finished eight in the Premier League last season - their highest ever finish.
Watch Sky Sports News from 9am this Wednesday for the announcement of next season's Premier League, Football League and SPFL fixtures.