The 24-year-old scored twice in 13 npower Championship appearances on loan at Pride Park this season and brings an end to his 15-year stay at Carrow Road.
Martin will join the Rams on July 1 having accepted a two-year deal with an option of a further year
He told Norwich's official website: "I am definitely going to look back on my time at Norwich with fond memories.
"There is a lot of good people there and a lot of friends I have made over time. There are a good set of lads there at the minute. I hope they stay up, kick on next season and do really well.
"It is a new start and a new beginning for me at Derby. There is a tinge of disappointment leaving Norwich behind, but I am very excited to be signing for another massive club in the form of Derby County."
Martin made his senior debut for Norwich in January 2007 and made 117 appearances, scoring 34 goals.