Saints and Scunthorpe had spent the last few days discussing a compensation package for the 45-year-old, who will replace Alan Pardew at St Mary's.
Pardew was sacked just two days after a 4-0 win at Bristol Rovers on August 28 with the club stating: "It is essential to make changes to the first team management and coaching".
Adkins, a former physio at Glanford Park before he became manager - twice leading them to promotion to the Championship - was always Southampton's first choice and he has now penned a three-year deal to work alongside trusted assistant Andy Crosby.
He said: "I am delighted to be the manager of Southampton.
"It's a very proud honour for myself, and Andy Crosby who is joining me as assistant manager, to join a club with such a big tradition.
"I'm very proud to have been given the opportunity by the chairman, Nicola Cortese, and look forward to taking the football club forward.
"I'm really looking forward to this fresh new challenge, and I'm very optimistic of where we can take this football club.
"We want to excite the supporters, because when this stadium is full I am sure there is a fantastic buzz around the place, so I hope the supporters are looking forward to watching their team go out and winning games of football in the right style."