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.
Championship club Norwich City have appointed Borussia Dortmund reserve-team coach Daniel Farke as their new head coach on a two-year deal.