Godfrey has signed a three-and-a-half year contract with the Premier League Canaries after the two clubs agreed an undisclosed fee.
"It's massive for me to be given this opportunity and I can't wait to get started," Godfrey told the club website.
"I'm delighted to get the deal done, and it happening on my birthday makes it extra special. York gave me the platform to kick on so I owe a lot to them, but it's about what happens next now."
Norwich boss Alex Neil confirmed that the teenager, who has made 15 senior appearances in his debut season for League Two outfit York, would initially be part of the club's Under-21 squad.