The 26-year-old spent the last four years with Gladbach and has also been capped by Switzerland 32 times and played in two World Cups.
Speaking to canaries.co.uk, Drmic said: "Now everything is done, I'm very happy. I had very good conversations with Stuart Webber and with the Head Coach Daniel Farke.
"I was talking also to Timm Klose [teammate with Switzerland]. We had a long conversation as I had a lot of questions and he spoke very, very positively about the Club.
"I'm very excited to be playing in the Premier League. 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."