The Republic of Ireland winger joins on a three-year contract for an undisclosed fee and is City’s third signing since promotion to the Premier League was secured last month.
The move means Brady leaves Hull after four seasons, two of them on loan, and 124 appearances.
“I’m really pleased to sign Robbie,” said Norwich boss Alex Neil, who has also signed Graham Dorrans and Youssouf Mulumbu this summer. “As everyone’s aware, we’ve been chasing him for quite a while.
“He’ll be an absolutely perfect fit for us. Hopefully he can go on and show everything that we think he’s capable of.
“He’s dynamic, he’s got good energy to get up and down the pitch, and he’s got a fantastic left foot which makes him a real danger at set-pieces. He’s got a lot of things in his locker.
“It gives us good competition down the left side. He’s got good pedigree and he’s still young so he’s got his best days ahead of him, so we’re just really happy that we’ve managed to get it over the line.”
Asked about the possibility of more signings, Neil told the club website: “We won’t be rushed. Any incomings have got to be right for the football club and right for the group.
“We want to make sure we do everything right and that we cover all bases, and that’s why this deal took a bit of time, but it’s important that we cover all that so we’re sure we make the right choices.”