The 24-year-old has passed a medical and signed a five-year deal at St Mary’s ending his two-year spell at Parkhead.
Van Dijk, who has been tracked by Southampton for the last 18 months, joined Celtic in 2013 after three years with Groningen and has won two successive league titles and the Scottish Cup with the Bhoys.
“I think it’s the right time for my development,” van Dijk told the club’s website. “I’m still quite young, 24, and still have a lot to learn.
“Southampton is the perfect step for me right now. It’s an amazing club. From the people I’ve met already it’s a very good feeling and I’m delighted to be here.
“I know it’s an ambitious club. They have been through a lot throughout the years. They are trying to play football and that’s something that really attracted me.
“I want to play as much as possible, get settled in quick and show everyone what I’m capable of and do well with the team.
“We are Southampton and we want to do better than last year. Hopefully that’s something that’s going to happen. I’m feeling good and I’m ready for everything.”
Meanwhile, Saints forward Emmanuel Mayuka, 24, has left to join Metz on a three-year deal.
The Zambia international, who was top scorer at the 2012 Africa Cup of Nations, joined Saints from Young Boys in the summer of 2012 and scored one goal in 19 appearances.
He spent the 2013-14 season on loan at French side Lorient and will return to France with Ligue 2 table toppers Metz.