The fee for Vestergaard is undisclosed, and the 25-year-old has signed a four-year deal with Saints according to the club.
Vestergaard made just under 200 Bundesliga appearances during his time at Hoffenheim, Werder Bremen and Mönchengladbach, gaining experience of European football in the process in both the Champions League and Europa League with the latter club.
The towering 6ft 6in defender was part of Denmark’s 23-man squad in the 2018 FIFA World Cup in Russia but didn’t feature as his country reached the last 16. Vestergaard has 16 senior caps for Denmark to his name.
Vestergaard is the fourth signing Saints has made in this summer transfer window so far following the arrivals of Stuart Armstrong from Celtic, Mohamed Elyounoussi from FC Basel and goalkeeper Angus Gunn from Manchester City.