Howson has agreed a three-and-a-half-year contract with the Barclays Premier League club.
The 23-year-old midfielder will be Norwich's first signing of the January transfer window.
The deal will go through officially once the paperwork has been approved "by the relevant footballing authorities", the Canaries confirmed.
Norwich manager Paul Lambert said: "I think he has been brilliant for Leeds - how he has played but also being their captain brings its own demands. It's a fantastic club Leeds.
"I think he can add goals to our midfield which I think we need and he can do that. He'll get better as a player the higher the level he goes up. If he does that and everything goes well, he'll be a great player for us."
Howson will wear shirt number 24 for Norwich. He has been out of action since December because of a knee injury suffered playing for Leeds against Millwall, but is hopeful of being fit soon.
The England Under-21 international had been mulling over signing a new deal with Leeds, but opted not to as they could not guarantee him top-flight football.
"The reason is Premier League," he told LUTV. "I am going to a very good club with good support, but at the same time I know I'm leaving a club I've loved from being a young boy. That's not going to change.
"Everyone at Leeds is wanting Premier League football. The ambition is always there, but that is not guaranteed."