Gunn impressed on loan at Norwich last season and was even drafted into England's squad to play Brazil last year, and also trained with the Three Lions ahead of this summer's World Cup.
The goalie could compete with Alex McCarthy and Fraser Forster for a starting spot at St Mary's next season, although their futures look uncertain following the signing of Gunn.
Saints manager Mark Hughes told Southampton's website: “Angus is a player who clearly possesses a tremendous amount of potential, but his performances for Norwich and for the England Under-21 team also demonstrate the capabilities he already has right now.
“We believe he has all the credentials to become a full international goalkeeper for England in the future, and that’s exactly the sort of talent we want to have here at Southampton.
“He represents another excellent addition, and we are excited about what he brings to the squad and the further progression we feel he can make with us.
“There was a lot of interest from other teams in the Premier League, so for Angus to choose Southampton as the best option for him is another positive indication of the work the club is doing and what is being built here.”
Gunn added: “The opportunity was there for me to stay at Manchester City but I saw the potential of this club and the squad.
“The manager and Dave Watson were a big part of the reason why I have chosen to join and speaking to them both was a big factor in my decision.
“I think the club has got a big future and I’m looking forward to getting started now. I’ve spoken to Dave Watson and he said it would be a great environment to come in to work. It feels really good and I’m buzzing to get going.”