The 28-year-old, capped six times by his country, made 50 appearances for Darmstadt after joining in 2015.
But the midfielder was unable to stop his side from being relegated from Germany's top flight this season and has moved Norwich on a three-year deal.
He will officially join on Friday, subject to governing bodies' approval.
Norwich head coach Daniel Farke, who himself moved from Germany to Carrow Road in May, told the club website: "I've followed Mario's path for several years.
"He's a really smart technical player with a brilliant left foot. He will bring a lot of quality passing and special creativity to the team.
"In addition to that, he's a really good character and a very nice guy who will fit into Norwich. I'm really looking forward to working with him."
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Norwich City have signed Bosnia international Mario Vrancic for an undisclosed fee from German side Darmstadt.