The 25-year-old has been strongly linked with a move to the Saints over the last couple of weeks, and the Premier League outfit have now confirmed his arrival on a deal until the summer of 2022.
Vestergaard made 34 appearances in all competitions for Gladbach during the 2017-18 campaign.
"We have had to fight off a lot of competition to bring Jannik here, and we are delighted he has chosen Southampton as the right place to take the next step in his career," Southampton boss Mark Hughes told the club's official website.
"Although we were aware of the level of interest in Jannik, he made it clear very quickly that the quality of the club and the ambitions we have here meant this was his number-one option.
"He represents another excellent addition to our squad this summer, and brings with him an extensive amount of experience for his age. That means he can not only help us in the present, but that he has the potential to develop his game even more with us, which is what we want as a club and is clearly exciting for both parties.
"One of our intentions this summer was to strengthen our aerial presence, and Jannik can help us do that not only defensively, but also in the opposition penalty area.
"His character and the leadership qualities he has shown in an intense division like the Bundesliga have been impressive, and, as a club, we have been following him for some time now, so we are naturally delighted to have been able to bring him to Southampton."