Danso, who was born to Ghanaian parents in Austria, but moved to England when he was six, came through the academy system at MK Dons, before moving to Augsburg in 2014.
He made his debut for the Bundesliga club's first-team in March 2017 and has gone on to make 45 appearances in total, while also establishing himself in the senior Austria side, earning six caps to date.
Danso said: “I'm delighted to join a club that has produced a lot of great players. I'm ready for the challenge and really excited to get started, and also to get to know the fans and the rest of the lads too.
“I'm looking forward to working with such a great coach in Ralph Hasenhüttl as well. He has produced a lot of young players and showed a huge amount of belief in them, so I hope I can be another one who can benefit from him.
“It's a great opportunity for me to show what I can do. Southampton is a great club, the manager is a great coach, and I'm really looking forward to the opportunity."
Southampton manager Ralph Hasenhüttl said: “I am very pleased we have been able to bring Kevin into the club.
“He fits the profile of player that we want to bring here to Southampton. He is in the early part of his career, but he has good experience already, with a lot of existing qualities and the potential to develop even further.
“We have made some good additions in our attack during the transfer window, with Ché Adams and Moussa Djenepo, and, like them, Kevin is another signing that fits our model of being a talented, young player who adds quality to our first-team squad. He will improve our defensive options, so this is an excellent way for us to finish our business."