Redmond, an England Under-21 international, has agreed a five-year contract with the Saints.
Norwich, who were relegated from the Premier League in May, signed the 22-year-old from Birmingham City for £3.2m in July 2013.
"They like to play attacking football and it's a style I feel I can really fit in well with," Redmond said.
"When Southampton said they were interested, it felt like the right move for me. I can develop my game here."
Birmingham-born Redmond made 122 appearances and scored 13 goals during his three-year spell with Norwich.
Redmond has played at every youth level for England, playing alongside Saints trio James Ward-Prowse, Jack Stephens and Matt Targett for England Under-21s at the Toulon Tournament earlier this summer.
Southampton are without a manager following Dutchman Ronald Koeman's departure to Everton.

The hosts declared on their overnight score of 244-5, setting Notts a victory target of 386 in 96 overs at The Oval.
Notts were 102-2 at lunch but collapsed spectacularly during the afternoon session, losing their final eight wickets in 53 minutes for just 38 runs.
Surrey spinner Zafar Ansari, who removed Jake Libby and Riki Wessels with successive balls, ended with 6-36.
Ansari's devastating spell meant Brendan Taylor's 68 from 79 deliveries was in vain for Notts, who are now level on points with Surrey in Division One.
The win also lifted Surrey off the bottom of the table above Hampshire.
Notts had looked in a good position while Taylor and opener Jake Libby, who made a patient 33 from 115 balls, were at the crease sharing a third-wicket stand of 90.
But the swift departures of Libby and Wessels started the procession of Notts wickets, with only Dan Christian (11) reaching double figures among their final seven batsmen.
Surrey captain Gareth Batty told BBC Radio London: "We've missed key moments in games at times, and we didn't miss them in this game.
"Everybody contributed and it ended up being a very good win against a very good Notts team.
"Traditionally, we're a better team in the second half of the season than in the first half. Hopefully we go back to type, play good cricket for the rest of the season and get a few more wins under the belt."

The 29-year-old was arrested on suspicion of attempted murder and wounding with intent.
Previously, a 30-year-old, 27-year-old and 20-year-old, all from Rugby, were arrested. All four men have been bailed until the end of April.
It follows an incident on Oxford Street on 17 March in which a 17-year-old was shot and an 18-year-old knifed.
More on this and other Warwickshire stories
Southampton have signed Norwich City winger Nathan Redmond for undisclosed fee believed to be worth £10m.
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Police investigating a shooting and stabbing in Rugby have arrested a fourth man.