The 26-year-old Drmic has penned a three-year contract having spent the last four seasons with Monchengladbach, and has UEFA Champions League experience from Bayer Leverkusen.
Drmic has 32 international caps, having represented Switzerland at two FIFA World Cups, and scored 10 goals for his country.
The Swiss is Norwich manager Daniel Farke’s second signing of the close season after the German sealed a season-long loan deal for Manchester City winger Patrick Roberts.
“I’m very excited to be playing in the Premier League," Drmic told the Norfolk club's official website.
"I cannot wait because I’ve heard a lot of things from my colleagues in the national team who have played in the Premier League.
“For me it’s something new. It’s a new opportunity with new challenges and I’m very happy that I’m so close now to playing in the Premier League. I cannot wait to start.”
Drmic began his career with FC Zurich before joining FC Nuremberg in 2013 where he finished his first season in the Bundesliga as third-top scorer with 17 goals.
He made his European debut with Leverkusen in 2014-15 before moving to Monchengladbach in June 2015.
Farke said: “We made it clear we wanted to sign a new striker and we’ve found a guy who has proved himself at the highest level.
“He has lots of Bundesliga appearances and has a very good record for the Switzerland national team."
Drmic scored just two goals in five Bundesliga games last season after battling a back injury and a muscle problem.
“He has had a difficult couple of years with injuries, but we feel he is the full package," said Farke.
"Two or three years ago, some of the biggest clubs in Europe were interested in him.
“He is a great character and a good guy, and we feel his best days are still to come.”
