The 19-year-old came through the Birmingham Academy and played under Canaries manager Chris Hughton when he was manager at St Andrew's, which included a run in the Europa League for the Championship club.
Having featured for England Under-21s in this summer's European Championship, Redmond is looking forward to testing himself in the top flight.
"Last season was a good one for me in terms of goals and assists and just general performances. Hopefully I can continue the form that I left off with last season at Birmingham," Redmond said on the official Norwich website, www.canaries.co.uk.
"Norwich looks like a fantastic club, I am just looking forward to getting started now."
Birmingham are understood to have also agreed a fee with Swansea for Redmond, in the region of £2million with add-ons. However, the winger opted instead to move to East Anglia and reunite with his former manager.
Redmond added: "Chris Hughton gave me the chance to play and the chance to express myself every time that I stepped on to the field, and I feel like on the bigger stage he will do exactly the same thing."
Redmond, though, admitted on a personal level it was not easy to be leaving his home-town club.
In a statement on his Facebook page, the 19-year-old wrote: "It has been a very emotional 11 years for me at Birmingham City FC and I have loved every single minute of it.
"I really, really am going to miss it from the bottom of my heart.
"My nan who passed away last year always used to tell me: 'Never forget where you come from.' Well I came from Birmingham City FC!! ONCE A BLUE ALWAYS A BLUE...
"This club will remain etched on my heart and soul for life, I am just sorry that I had to leave, but I am looking forward to the next step on my journey as a professional footballer .
"Thank you so much to you fans for supporting me on my journey from schoolboy to young professional."
Redmond becomes Norwich's third signing of the summer, following the permanent deal for on-loan Lazio defender Javier Garrido and the club-record £8.5m transfer of Holland forward Ricky van Wolfswinkel.
Hughton is in no doubt there is more to come from Redmond.
"He is very keen to develop his own game, so I am sure the move won't faze him, but this is still a part of his development and one he is very much looking forward to," the Norwich manager said.
Hughton is keen to further bolster his squad ahead of their three-match pre-season tour of the United States, and has been linked with a swoop for Sweden international Ola Toivonen of PSV Eindhoven.
However, earlier this week the Norwich manager maintained the club would resist any moves to lure away England goalkeeper John Ruddy, with Chelsea ready to significantly improve their £5million offer which was swiftly rejected.