The 24-year-old joined the Canaries from Leeds United for an undisclosed fee and believes he can help establish Norwich as a Premier League force.
Snodgrass told the club's official website: "I'm delighted to get it done and dusted and now I'm looking forward to meeting my new team-mates and starting the season. Sometimes this window of opportunity doesn't come about too often, and I jumped at the chance.
"I firmly believe that the Premier League is the best league in the world, and when Norwich came in for me, I made the step because I believe that Norwich can kick on again."
The Scotland international will be in contention to make his Canaries debut in their Premier League opener against Fulham on August 18.