Tudor-Jones, 24, has made a total of 56 appearances for the Swans and netted four times from Swansea.
Gunn told the club's official website: "After Owain had figured for Wales in their recent win over Azerbaijan we sat down with the player and his agent to thrash out terms and I'm delighted we are able to confirm we have brought him in on a three-year deal."
Tudur-Jones said: "I'm looking forward to the next few years that I'll be spending at the club. Having been and had a look at the facilities it's a club I'm throughly looking forward to playing for and all the challenges that lie ahead. I've spoken to the manager and listened to his plans for building the club back up and I want to be a part of that."
Meanwhile, Ian Butterworth is delighted Norwich have been given a home game against Colchester on August 8 to start their League One campaign.
Butterworth is upbeat about the season's opening fixture and said: "We're glad to be home against Colchester. It's probably the nearest thing we are going to get to a derby game, so we are all going to be looking forward to that.
"We've got to start positively and hit the ground running. We've got to put on a good show in front of our fans - it is as simple as that."