The former Scotland international joins on a three-year deal after leaving Swansea.
Martin verbally agreed to take over at St Mary’s last month, while the clubs remained in discussions over compensation.
“It’s a privilege to accept this opportunity at Southampton, a club with such a long and rich history,” he said.
“My aim is to get this club back where it belongs – in the top flight of English football. I am ready for this challenge and will give everything to achieve this aim and to give the supporters a team that makes them proud. I’m thrilled to be here and cannot wait for the work to begin.”
Leicester were also interested in the 37-year-old but Southampton moved quickly following their relegation from the Premier League.
Several of his backroom staff, including assistant Mat Gil, are expected to join Martin.
Southampton and Swansea disagreed over the amount of compensation which should be paid, which has seen the move dragged out.
They had been haggling over £700,000 with Martin’s release clause for Championship clubs and Premier League sides different.
The Saints felt they should pay the smaller amount, around £1.25million, after their relegation from the top flight while the Swans were believed to be holding out for £2million.
Martin guided Swansea to 10th in the Championship last season despite having a bottom-four budget and the youngest squad in the division.
He has been praised for his style and Southampton will be hoping the ex-Norwich defender can guide them back to the Premier League at the first attempt.
They finished bottom after a disastrous season which has seen them use three managers.
Nathan Jones replaced Ralph Hasenhuttl in November but lasted just 14 games, winning one, and was sacked in February. Ruben Selles took charge for the final months of the campaign but was unable to keep Saints up.
Last month chief executive Martin Semmens stepped down from his role following relegation having been in the post since 2019.
Jason Wilcox was announced as Southampton’s director of football in January, joining from Manchester City, and will officially start work next week.
Wilcox added: “Russell has shown fantastic qualities in his managerial career so far and is someone who we believe will thrive in the great environment we have here at Southampton.
“He is a fantastic fit for what we are trying to achieve, with a strong record of developing and nurturing young, talented players to fulfil their potential and deliver results on the pitch. We can’t wait for Russell to get started working with the team at Staplewood as we get ready for the new season.”
Swansea are in talks with Barnsley boss Michael Duff, as revealed by the PA news agency on Sunday, with the former Burnley midfielder expected to be Martin’s replacement.
More from Sporting Life
Fast Results
Scores
