The 24-year-old is the third summer signing for manager Chris Hughton after Jacob Butterfield and Steven Whittaker.
The Scotland international links up with former Leeds team-mates Jonny Howson and Bradley Johnson at Carrow Road.
Snodgrass told Norwich's website: "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.
"Sometimes this window of opportunity doesn't come about too often, and I jumped at the chance.
"What Norwich did in the Premier League last season speaks for itself.
Bradley Johnson and Jonny Howson kicked on for Leeds, and when you look at that, you think 'I want to be part of it' and try to get Norwich as far up the table as they can possibly go."
Snodgrass joined Leeds in 2008, playing 168 league matches for United, scoring 35 times.
He was named the 2011/12 Player of the Year and Player's Player of the Year, as well as being given the captain's armband upon Neil Warnock's arrival at the helm in February 2012.
Canaries boss Chris Hughton said: "Robert is an excellent addition to the squad and I'm delighted to add a player of his calibre to the team.
"He is a player I have admired for a number of years now and he's at a good age at 24, with some excellent experience from his time with Leeds.
"He's also got that international experience with Scotland and is used to playing in big matches both with his country and at Elland Road."