The 40-year-old German replaces Alex Neil, who left the club in March.
Farke, who spent his playing career in Germany's lower leagues, replaced Huddersfield boss David Wagner at Dortmund's second team in 2015.
Norwich's sporting director Stuart Webber was part of the Terriers' backroom team when Wagner joined.
Edmund Riemer, Farke's assistant at Dortmund, has been appointed assistant head coach at Carrow Road.
"I had a really good time with Borussia Dortmund, but Stuart Webber has told me all about Norwich City, its traditions and history, and I know what a great club it is," Farke told the club website.
"The more I know about Norwich City, the more excited I get. I'm very glad to be here and can't wait to get started."
Norwich finished eighth in the Championship table this season, 10 points outside the play-off places.

Cheng says a lot of people who aren't funny are afraid to be "weird" or "deviate from what's normal".
This is a mistake, he says, arguing that comedians shouldn't be worried about having controversial views.
"I think a lot of funniness comes from ideas which may be different, and that's why a lot of comedians come from a different background and have a different point of view," he says.
"Embrace that."
Cheng himself dropped out of his maths degree at Cambridge University in 2008 to become a professional poker player before later getting into comedy.
He brings an analytical approach to his comedy, using graphs and a PowerPoint presentation in his Edinburgh performance.
Read about the joke that won him the award, here.
With the growth of online streaming sites like Netflix, there is a lot of comedy available.
Cheng says those wanting to be funnier should be watching plenty of it - particularly the things they find most funny.
Doing this helps develop a sense of humour and potentially a distinct comedic voice, Cheng says.
"Start off imitating it and you will find your own voice, because a lot of comedy comes by imitating enough things, which can create a different voice."
Making friends laugh doesn't only make you and them feel good - it can be a stepping stone to making others laugh, Cheng argues.
"If you make your friends laugh I think it's one of the greatest ways to connect with people. It's a stepping stone to making others laugh."
While Cheng didn't do it himself, he knows other comedians and friends who have taken stand-up and improvisational comedy courses.
These courses can give people an understanding of joke writing and experience of performing in a supportive environment.
Cheng developed his own comedy skills in the familiar environment of students gigs in Cambridge, where he continued to live after dropping out of his degree in 2008.
Cheng says social media can be a good place to try out jokes without the pressure of getting on stage.
"It's a good way, because you don't have to go in front of people and you can realise 'I know how to do this.'"
But above all else, Cheng has some words of encouragement for people who hope to be funny.
"If you have a sense of humour then you probably have the potential to be funny," he adds.
"Being a Chinese comedian is much easier than being a female comedian because no one ever comes up to you after a gig and says 'you were pretty funny... for a Chinese comedian'."
"I walked in on my flatmate looking at an optical illusion with his pants down. He jumped up and screamed 'it's not what it looks like!'"
Ken Cheng is performing at The Cellar, Pleasance Courtyard, at the Fringe Festival, each day at 16:45 BST until 27 August.
Championship club Norwich City have appointed Borussia Dortmund reserve-team coach Daniel Farke as their new head coach on a two-year deal.
[NEXT_CONCEPT]
The winner of the best joke award at this year's Edinburgh Festival Fringe, Ken Cheng, shares five tips on how to be funnier, be it on stage or just in front of a group of friends.