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, where he made 117 appearances, netting 34 goals.
Martin will join the Rams on July 1 having accepted a two-year deal with an option of a further year and becomes boss Nigel Clough's second signing this week after bringing in goalkeeper Lee Grant from Burnley.
Clough told his club's official website: "Chris did really well whilst here on loan so we're pleased to get him signed up and it's another important signing for us.
"He's a striker with different qualities to what we've already got, especially with his physical presence and ability to hold the ball up.
"We've admired him from afar for some time and monitored his progress closely and once we got him in the building we knew he would fit in just fine with how we want to play and this group of lads.
"We can't wait to see him in action come the start August with a full pre-season behind him."