Drmic joins on free transfer and has penned a three-year deal at Carrow Road, after four seasons with Gladbach.
The Switzerland striker is Norwich's second new arrival ahead of the 2019-2020 season after Manchester City winger Patrick Roberts joined on loan.
"I'm coming to the Premier League and I can now challenge in the best league in the world," Drmic said. "I'm going to do everything and give 100 per cent on the pitch. I will be ready to give everything for the club.
"My job is scoring but I also want to help my team and help us be successful.
"When I first came to Norwich, the first thing I noticed was how kind everybody was. It's given me a lot of positive energy and I'm excited to see what happens."
Drmic, who has also played for Nurnberg, Bayer Leverkusen and Hamburg, scored only seven Bundesliga goals for Gladbach during his time at the club.
"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," Norwich manager Daniel Farke said. "He has lots of Bundesliga appearances and has a very good record for the Switzerland national team.
"He has had a difficult couple of years with injuries, but we feel he is the full package. 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."